Existence of regular nut graphs for degree at most 11
arXiv:1908.11635v2 [math.CO] 6 Nov 2019
Patrick W. Fowler∗, John Baptist Gauci†, Jan Goedgebeur‡§,
Tomaž Pisanski¶, and Irene Scirihak
Dedicated to the memory of Slobodan Simić.
Abstract
A nut graph is a singular graph with one-dimensional kernel and corresponding eigenvector
with no zero elements. The problem of determining the orders n for which d-regular nut graphs
exist was recently posed by Gauci, Pisanski and Sciriha. These orders are known for d ≤ 4. Here
we solve the problem for all remaining cases d ≤ 11 and determine the complete lists of all dregular nut graphs of order n for small values of d and n. The existence or non-existence of small
regular nut graphs is determined by a computer search. The main tool is a construction that
produces, for any d-regular nut graph of order n, another d-regular nut graph of order n + 2d. If
we are given a sufficient number of d-regular nut graphs of consecutive orders, called seed graphs,
this construction may be applied in such a way that the existence of all d-regular nut graphs of
higher orders is established. For even d the orders n are indeed consecutive, while for odd d the
orders n are consecutive even numbers. Furthermore, necessary conditions for combinations of
order and degree for vertex-transitive nut graphs are derived.
1
Introduction
In this paper we consider graphs that are simple, that is, without loops or multiple edges. Given a
labelled graph G of order n, its 0 − 1 adjacency matrix A = A(G) is the n × n matrix with entries
aij (for i, j ∈ {1, . . . , n}) such that aij = 1 if and only if there is an edge between the vertices i and
j in G. If A(G) has an eigenvalue zero, the graph G is singular with nullity η = η(G) equal to the
multiplicity of the eigenvalue zero of A(G). In the sequel, we use terminology for a graph and its
adjacency matrix interchangeably. A graph that has an all-non-zero eigenvector corresponding to the
zero eigenvalue is a core graph. If, in addition, the dimension of the nullspace is one, the corresponding
graph is a nut graph [10, 11, 12, 13, 14]. Some authors consider the isolated vertex to be a trivial case
of a nut graph, and would consider nuts proper, or non-trivial nuts, to have n > 1. In the present
paper, the only nut graphs that are considered are non-trivial, in this sense.
Two chemical motivations can be claimed for the study of nut graphs of maximum degree ≤ 3 (the
chemical nut graphs). First, the eigenvalue zero is special in a chemical context, as it corresponds in
Hückel Molecular Orbital Theory to the non-bonding energy level of a conjugated carbon π system,
from which electrons are easily removed and to which they are easily added. If this happens to be the
highest occupied molecular orbital, then partial occupation will correspond to a radical (molecule with
unpaired spins) in which spin density and hence radical reactivity is distributed over all carbon atoms
∗ Department
of Chemistry, University of Sheffield, Sheffield, United Kingdom (p.w.fowler@sheffield.ac.uk)
of Mathematics, Faculty of Science, University of Malta, Msida, Malta (john-baptist.gauci@um.edu.mt)
‡ Department of Applied Mathematics, Computer Science & Statistics, Ghent University, Ghent, Belgium
(jan.goedgebeur@ugent.be)
§ Computer Science Department, University of Mons, Mons, Belgium
¶ Department of Mathematics, IMFM, University of Ljubljana and University of Pri- morska, Ljubljana, Slovenia
(pisanski@upr.si)
k Department of Mathematics,
Faculty of Science, University of Malta, Msida, Malta (irene.scirihaaquilina@um.edu.mt)
† Department
1
of the framework. Secondly, it has been shown [5] that nut graphs correspond exactly to the classes
of distinct and strong omni–conductors of nullity one, according to the simplest model of molecular
conduction. (A distinct omni–conductor is a molecule for which connection in a circuit via any distinct
pair of atoms gives transmission; for a strong omni–conductor, the transmission occurs whether the
connection atoms are distinct or not.)
Using a computer search, Fowler, Pickup, Todorova, Borg and Sciriha [5] determined all nut
graphs up to 10 vertices, and all chemical nut graphs up to 16 vertices. Later Coolsaet, Fowler and
Goedgebeur [4] designed a specialised algorithm to generate nut graphs and used it to determine all
nut graphs up to 13 vertices and all chemical nut graphs up to 22 vertices. We refer the reader to [4]
and [6] for a more detailed overview.
In [14] Sciriha and Gutman proved that the smallest non-trivial nut graphs have order 7 and that
nut graphs exist for all orders at least 7. A natural question arises:
Question 1. What is the smallest order for which a regular nut graph exists?
A chemical motivation for the study of regular nut graphs is that 3-regular, 3-connected graphs
are candidates as carbon cages, so that nut graphs of this type would themselves be candidates for
omni-conductors and radical molecules of the kinds mentioned above.
The two main results of [6] can be stated as follows.
Theorem 1. Cubic nut graphs on n vertices exist if and only if n is an even integer, n ≥ 12 and
n∈
/ {14, 16}.
Theorem 2. Quartic nut graphs on n vertices exist if and only if n = 8, 10, 12 or n ≥ 14.
The results of a computer search presented in Figure 2 have indeed shown that the answer to the
above question is the following.
Remark 1 (Answer to Question 1). The smallest order for which a regular nut graph exists is 8.
Since no nut d-regular graph exists for d = 1 or d = 2, the above results raise a number of related
questions. (See also Section 4 “Conclusion”).
Question 2. Is it true, that for each d > 2 there exists a d-regular nut graph?
In [6], the set N (d) was defined as the set consisting of all integers n for which a d-regular nut
graph of order n exists. Therefore we have N (1) = N (2) = ∅, N (3) = {12} ∪ {2k : k ≥ 9},
N (4) = {8, 10, 12} ∪ {k : k ≥ 14}.
Definition 1. Let us call a pair (n, d) admissible if there exists a d-regular (simple) graph of order n
and let us denote by A(d) the collection of integers n, such that there exists an (n, d)-admissible graph.
Let us call a pair (n, d) nut-realisable if and only if there exists a d-regular nut graph of order n.
For d even, (n, d) is admissible if and only if d < n. For d odd, (n, d) is admissible if and only if
d < n and n is even. Clearly N (d) ⊂ A(d). A valence d for which the set A(d) \ N (d) is finite will be
called normal.
Question 3. Is it true, that for each d > 2 there are only a finite number of admissible orders that
do not belong to N (d)? This is equivalent to saying, is the complement A(d) \ N (d) finite? In other
words, is it true that d is normal if and only if d > 2?
In this note, we determine N (d), and give positive answer to the above question, for every d ≤ 11.
That is:
Theorem 3. The following holds:
• N (5) = {2k : k ≥ 5}
2
• N (6) = {k : k ≥ 12}
• N (7) = {2k : k ≥ 6}
• N (8) = {12} ∪ {k : k ≥ 14}
• N (9) = {2k : k ≥ 8}
• N (10) = {k : k ≥ 15}
• N (11) = {2k : k ≥ 8}
We use a construction that was proved and applied repeatedly in [6]. It extends any singular graph
G at a vertex v of valency d, to give a graph F (G, v) with the same nullity as G but with 2d more
vertices (each of degree d), whilst all other vertices retain the degree they had in G. The process used
in this extension, named in [6] the Fowler Construction, is depicted in Figure 1.
(1 − d)xv
xv
⇒
u1
u2
...
ud
q1
q2
...
qd
p1
p2
...
pd
u1
u2
...
ud
F (G, v)
G
Figure 1: The ‘Fowler Construction’ [6]. Notation: xv , ui are entries in the kernel eigenvector of G, with
entries ui summing to zero; in F (G, v), the two layers of extra vertices span a complete bipartite
graph minus a perfect matching, with eigenvector entries qi = ui , pi = xv , and the entry on vertex
v replaced by (1 − d)xv .
From the construction, it follows immediately that:
Lemma 4. Let G be a nut graph on n vertices and let v be a vertex of G having valency d. Then
there exists a nut graph F (G, v) on n + 2d vertices in which the 2d new vertices all have valency d.
Starting from a nut graph of order n, the above lemma can be used repeatedly to produce an
infinite sequence of nut graphs on n, n + 2d, n + 4d, n + 6d, . . . vertices in which all of the newly
introduced vertices have valency d. In particular, this argument can be applied to any d-regular nut
graph on n vertices to obtain an infinite sequence of d-regular nut graphs.
For even valency d, one has to establish the existence of d-regular nut graphs for 2d consecutive
orders, starting with order n. By Lemma 4 above, this would then imply the existence of d-regular
nut graphs of any order at least n. Smaller cases have to be checked by computer search or some
other method. In the case of d = 4, such a run of eight consecutive orders was obtained starting at
n = 14 [6]
By the well-known Handshaking Lemma, for odd valency d the order of a graph must be even. It
therefore suffices to establish existence of d-regular nut graphs for d consecutive even orders, starting
with order n. For d = 3 the run of three even orders started at n = 18 [6].
3
2
Existence of regular nut graphs
In [4], a generation algorithm for nut graphs is presented. We could in principle use this generator
and restrict the generation to d-regular graphs. However, it turned out to be more efficient to use
Meringer’s generator for regular graphs, genreg [9], to generate d-regular graphs and then test which
are nut graphs (using the filter program from [3]).
Using this approach, we determined all 5-regular nut graphs up to 18 vertices. The counts can
be found in Table 1. Using a similar approach (but using the generator from [2] for the cubic case),
we also determined all 4-regular nut graphs up to 15 vertices and all 3-regular nut graphs up to 22
vertices, the counts for which can be found in Table 1. These graphs can also be downloaded from
the House of Graphs [1] at https://hog.grinvin.org/Nuts.
We now proceed to the statement and proof of our main result.
Theorem 3. The following holds:
• N (5) = {2k : k ≥ 5}
• N (6) = {k : k ≥ 12}
• N (7) = {2k : k ≥ 6}
• N (8) = {12} ∪ {k : k ≥ 14}
• N (9) = {2k : k ≥ 8}
• N (10) = {k : k ≥ 15}
• N (11) = {2k : k ≥ 8}
Proof. It follows from Table 1 that the smallest 5-regular nut graphs have 10 vertices. In order to
cover all orders using the construction, we need to present a 5-regular nut graph of every even order
10 ≤ n ≤ 18. In the Appendix we give an example of a 5-regular nut graph of every such order n.
In fact, the examples from the Appendix are chosen to exhibit the largest automorphism group size
amongst the 5-regular nut graphs of that order.
The proof for the other cases of N (d) for 6 ≤ d ≤ 11 is completely analogous. That is: using
Meringer’s generator genreg [9] together with a filter which tests whether a given graph is a nut, we
determined the smallest order for which a d-regular nut graph exists and excluded small orders for
which d-regular nut graphs do not exist (e.g. to prove the nonexistence of an 8-regular nut graph on
13 vertices). See Table 1 for the counts of the smallest d-regular nuts for 6 ≤ d ≤ 11. We have used
genreg to establish the existence of a d-regular nut graph of order n for 2d consecutive orders (if d is
even), or the existence of a d-regular nut graph for d consecutive even orders (if d is odd) which are
then used as seeds for the construction.
The adjacency list of a d-regular seed graph of every required order can be found in the Appendix.
These graphs can also be inspected in the database of interesting graphs from the House of Graphs [1]
by searching with keywords “d-regular nut graph” (with d ∈ {5, . . . , 11}).
3
Vertex-transitive nut graphs
A graph G such that all vertices are equivalent under the automorphism group Aut(G) is vertextransitive. Requiring regular nut graphs to be vertex-transitive clearly imposes further restrictions,
and it seems natural to ask the following question.
Question 4. For what pairs (n, d) does a vertex-transitive nut graph of order n and degree d exist?
4
3-regular nut graphs
Number of
Order
graphs
12
9
18
5 541
20
5
22
71
4-regular nut graphs
Number of
Order
graphs
8
1
10
12
12
269
14
15 633
15
1
5-regular nut graphs
Number of
Order
graphs
10
9
12
4
14
25
16
13 530
18
665 456 900
6-regular nut graphs
Number of
Order
graphs
12
1 964
13
79
14
1 872
7-regular nut graphs
Number of
Order
graphs
12
3
14
5 168 453
8-regular nut graphs
Number of
Order
graphs
12
24
13
0
14
424 088
9-regular nut graphs
Number of
Order
graphs
14
0
16
>0
10-regular nut graphs
Number of
Order
graphs
14
0
15
173 650
11-regular nut graphs
Number of
Order
graphs
14
0
16
3 316
Table 1: The numbers of d-regular nut graphs for d ∈ {3, . . . , 11}. No such graphs exist for smaller orders.
Some straightforward observations give the following necessary conditions for the existence of a
vertex-transitive nut graph, as summarised in the following theorem:
Theorem 5. Let G be a vertex-transitive nut graph on n vertices, of degree d. Then n and d satisfy
the following conditions. Either d = 0 mod 4, and n = 0 mod 2 and n ≥ d + 4; or d = 2 mod 4, and
n = 0 mod 4 and n ≥ d + 6.
Proof. We are going to prove the result in five steps.
Let x be a kernel eigenvector of a vertex-transitive nut graph G. As the zero eigenvalue is simple,
x transforms as a non-degenerate irreducible representation of Aut(G). Hence, the eigenvector can be
chosen to have all entries real. Under any given automorphism, each eigenvector entry xi transforms
to xj or −xj for some j, and therefore all squared entries x2i are equal. We can choose a normalisation
where each entry in x is either +1 or −1.
A kernel eigenvector obeys the local adjacency condition:
X
xj = 0
(1)
i∼j
for every vertex i, which implies that the neighbourhood of every vertex of a vertex-transitive nut
graph contains an equal number of vertices bearing entries +1 and −1 in x. Hence, a vertex-transitive
nut graph is of even degree.
Next, note that a kernel eigenvector x of a d-regular graph obeys
n
X
xi = 0,
(2)
i=1
as x is orthogonal to the all-ones Perron eigenvector that corresponds to the maximum eigenvalue
λmax = d of a regular graph. Hence, for a vertex-transitive nut graph G, the sum over all of the +1
and −1 entries xi is zero, from which we deduce that n is even.
5
Consider the subgraphs of G induced by the vertices with entries xi = +1 and xi = −1, respectively.
Let H be the subgraph induced by the vertices with xi = +1. Vertex i in G is of even degree, and
in the neighbourhood of i in G there are d/2 vertices carrying entries of each sign. The graph H is
therefore regular, of degree d/2 and of order n/2. Hence, if d = 2 mod 4, d/2 is odd and the order of
H must be even. Therefore, n/2 is even and we have n = 0 mod 4 for d = 2 mod 4.
Finally, note a limitation on the smallest vertex-transitive nut graph of given degree d. A d-regular
graph has at least (d + 1) vertices. Given that both degree and order of a vertex-transitive nut graph
are even, the order of a vertex-transitive nut graph obeys n ≥ d + 2. However, a vertex-transitive nut
graph with n = d + 2 is not possible. For even n and d, there is a unique vertex-transitive graph, H ′ ,
of order d + 2 [8], which is not a nut graph. Indeed, as every vertex of H ′ is adjacent to all but one
of the others, we can choose a labelling such that vertices i and d + 1 − i of H ′ are duplicates for all
0 ≤ i ≤ n − 1. In particular, 0 and d + 1 have the same neighbourhood, {1, . . . , d}. The subgraph
induced by this neighbourhood is Kd minus a perfect matching, as edges between j and d + 1 − j are
missing for 1 ≤ j ≤ d. This is the well known cocktail-party graph on d vertices, CP (d/2). It has
d/2 distinct pairs of non-adjacent vertices. A kernel eigenvector for H ′ can be constructed as follows:
assign values +1 to vertex 0, −1 to vertex d + 1, and zero to all others. Hence, H ′ is not a nut graph,
and the stated limits on n follow.
Calculations using the catalogue of vertex-transitive graphs created by Holt and Royle [7] indicate
that all pairs (n, d) compatible with the above conditions and with n ≤ 42 correspond to at least one
vertex-transitive nut graph.
See, for example, the blue entries in Figure 2. This figure gives an overview of the existence of
d-regular and vertex-transitive nut graphs of order n ≤ 30.
Our question is therefore refined to:
Question 5. For what pairs (n, d) obeying Theorem 5 does a vertex-transitive nut graph of order
n and degree d exist? (In other words: what pairs are realisable, in the sense that a pair (n, d) is
realisable if there exists at least one vertex-transitive nut graph with these parameters?)
We do know that there is at least one infinite family of realisable pairs (n, d) of vertex-transitive
graphs, as all antiprisms with largest ring size not divisible by 3 are nut graphs; hence all pairs (n, 4)
with n 6= 0 mod 6 are vertex-transitive-nut-realisable.
4
Conclusion
We divide the set of nut graphs into two disjoint subsets. If a graph can be obtained from the Fowler
construction, we call it a C-graph. If it cannot be obtained in this way, we call it a seed graph or
S-graph. This carries over to nut-realisable pairs (n, d). Such a pair is a C-pair if there exists at least
one (n, d)-graph that is a C-graph. On the other hand, if all (n, d)-graphs are seed graphs, we call the
pair (n, d) an S-pair or seed pair.
Proposition 6. If d is a normal valence, then the following is true. If d is odd there are exactly d
seed pairs (n, d) and if d is even, there are exactly 2d such seed pairs. In the former case each seed
value of n has a different (even) remainder mod 2d while in the latter case each n gives a different
remainder mod 2d.
Let d be a normal valence. We can define two numbers: n0 (d) and n1 (d). Here n0 (d) represents
the smallest value of n such that the pair (n, d) is nut-realisable. On the other hand let n1 (d) represent
the smallest value such that admissible pairs (n, d) with n ≥ n1 (d) are nut-realisable.
Our results from Figure 2 lead to the following values of n0 (d) and n1 (d) in Table 2.
Variants of the original question suggest themselves. For example:
Question 6. Determine the set N (p, g) such that there exists a p-regular nut graph of order n and
and girth (at least) g if and only if n ∈ N (p, g).
6
n
7
8
9
10
11
X
X
S
X
12
13
14
15
16
17
18
19
20
X
X
S
S
X
C
S
C
S
C
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
21
22
23
S
C
S
C
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
X
S
S
24
25
26
27
28
29
30
d
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
X
X
S
X
X
S
S
X
X
X
X
S
S
X
S
S
S
X
S
X
X
X
X
S
S
S
S
X
X
X
X
C
S
S
S
S
S
S
S
S
S
S
X
S
X
X
S
C
X
X
X
S
S
S
S
S
S
S
X
S
S
S
S
S
S
X
S
X
X
X
X
C
C
C
C
C
C
C
C
C
C
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
C
C
S
S
S
S
S
S
S
S
S
S
S
S
S
S
X
S
X
X
C
C
?
X
C
C
C
C
C
C
C
C
C
S
C
S
S
S
S
C
C
C
C
S
S
S
S
S
S
S
S
S
S
S
S
S
S
?
S
S
S
?
S
S
?
?
S
?
X
X
X
S
?
S
?
S
X
X
S
?
?
S
?
S
S
?
?
X
?
?
S
?
S
X
X
Figure 2: Snapshot of small regular nut graphs: calculations and proven results for order n and degree d for
n ≤ 30. Symbols: blank spaces and red crosses indicate non-existence; question marks indicate
open cases; the symbol S indicates a seed graph; the symbol C indicates that at least one example
of a regular nut graph can be obtained by the Fowler construction; blue (resp., green) symbols
indicate the existence (resp., non-existence) of a vertex-transitive nut graph.
Question 7. Determine the set N (p) such that there exists a q-regular nut graph of order n for all
degrees 3 ≤ q ≤ p if and only if n ∈ N (p).
Acknowledgements
Jan Goedgebeur is supported by a Postdoctoral Fellowship of the Research Foundation Flanders
(FWO). The work of Tomaž Pisanski is supported in part by the Slovenian Research Agency (research
program P1-0294 and research projects N1-0032, J1-9187, J1-1690), and in part by H2020 Teaming
InnoRenew CoE. Several computations for this work were carried out using the Stevin Supercomputer
Infrastructure at Ghent University.
References
[1] G. Brinkmann, K. Coolsaet, J. Goedgebeur, and H. Mélot. House of Graphs: a database of
interesting graphs. Discrete Applied Mathematics, 161(1-2):311–314, 2013. Available at https:
//hog.grinvin.org/.
7
d
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
n0 (d)
12
8
10
12
12
12
16
15
16
16
18
19
20
20
22
22
24
24
≥ 26
26
≥ 28
28
≥ 30
30
n1 (d)
18
14
10
12
12
14
16
15
16
≥ 16
≥ 18
≥ 19
≥ 20
≥ 20
≥ 22
≥ 22
≥ 24
≥ 24
≥ 26
≥ 26
≥ 28
≥ 28
≥ 30
≥ 30
Table 2: Known values of n0 (d) and n1 (d). An entry of the form n0 (d) ≥ p, indicates that a complete search
of all feasible smaller cases of order less than p has not revealed an example of a d-regular nut graph.
[2] G. Brinkmann, J. Goedgebeur, and B.D. McKay. Generation of cubic graphs. Discrete Mathematics and Theoretical Computer Science, 13(2):69–80, 2011.
[3] K. Coolsaet, P.W. Fowler, and J. Goedgebeur. Homepage of Nutgen: http://caagt.ugent.be/
nutgen/.
[4] K. Coolsaet, P.W. Fowler, and J. Goedgebeur. Generation and properties of nut graphs. MATCH
Commun. Math. Comput. Chem., 80(2):423–444, 2018.
[5] P.W. Fowler, B.T. Pickup, T.Z. Todorova, M. Borg, and I. Sciriha. Omni-conducting and omniinsulating molecules. The Journal of Chemical Physics, 140(5):054115, 2014.
[6] J.B. Gauci, T. Pisanski, and I. Sciriha. Existence of regular nut graphs and the Fowler construction. arXiv preprint arXiv:1904.02229, 2019.
[7] D. Holt and G.F. Royle. A census of small transitive groups and vertex-transitive graphs. Journal
of Symbolic Computation, 2019. To appear. DOI: https://doi.org/10.1016/j.jsc.2019.06.
006.
[8] B.D. McKay and G.F. Royle. The transitive graphs with at most 26 vertices. Ars Combinatoria,
30:161–176, 1990.
[9] M. Meringer. Fast generation of regular graphs and construction of cages. Journal of Graph
Theory, 30(2):137–146, 1999.
8
[10] I. Sciriha. On the construction of graphs of nullity one. Discrete Mathematics, 181(1):193–211,
1998.
[11] I. Sciriha. On the rank of graphs. In Graph Theory and Algorithms in ed Y. Alavi, D.R. Lick
and A. Schwenk, volume II, pages 769–778, 1999.
[12] I. Sciriha. A characterization of singular graphs. Electronic Journal of Linear Algebra, 16:451–462,
2007.
[13] I. Sciriha. Coalesced and embedded nut graphs in singular graphs. Ars Mathematica Contemporanea, 1:20–31, 2008.
[14] I. Sciriha and I. Gutman. Nut graphs: maximally extending cores. Utilitas Mathematica, pages
257–272, 1998.
Appendix
Adjacency lists of regular nut graphs
The graphs in this section act as seed graphs in the proof of Theorem 3. For completeness we also
list the seed graphs for d = 3 and d = 4. These graphs can also be inspected at the database of
interesting graphs from the House of Graphs [1] by searching for the keywords “d-regular nut graph”
(with d ∈ {3, ..., 11}).
Adjacency lists of 3-regular nut graphs
• Order 12: {0: 1 2 3; 1: 0 4 5; 2: 0 6 7; 3: 0 8 9; 4: 1 6 8; 5: 1 7 9; 6: 2 4 10; 7: 2 5 11; 8: 3 4 11; 9: 3 5 10; 10: 6
9 11; 11: 7 8 10}
• Order 20: {0: 1 2 3; 1: 0 4 5; 2: 0 6 7; 3: 0 8 9; 4: 1 6 10; 5: 1 8 11; 6: 2 4 12; 7: 2 11 13; 8: 3 5 14; 9: 3 12 15;
10: 4 13 16; 11: 5 7 17; 12: 6 9 16; 13: 7 10 18; 14: 8 15 19; 15: 9 14 18; 16: 10 12 19; 17: 11 18 19; 18: 13 15
17; 19: 14 16 17}
• Order 22: {0: 9 14 15; 1: 10 16 18; 2: 11 13 14; 3: 11 13 15; 4: 12 18 21; 5: 12 20 21; 6: 15 16 17; 7: 16 19 20; 8:
17 19 20; 9: 0 14 19; 10: 1 17 18; 11: 2 3 13; 12: 4 5 21; 13: 2 3 11; 14: 0 2 9; 15: 0 3 6; 16: 1 6 7; 17: 6 8 10;
18: 1 4 10; 19: 7 8 9; 20: 5 7 8; 21: 4 5 12}
Adjacency lists of 4-regular nut graphs
• Order 8: {0: 1 2 3 4; 1: 0 2 3 5; 2: 0 1 4 6; 3: 0 1 5 7; 4: 0 2 6 7; 5: 1 3 6 7; 6: 2 4 5 7; 7: 3 4 5 6}
• Order 10: {0: 1 2 3 4; 1: 0 2 3 5; 2: 0 1 4 6; 3: 0 1 5 7; 4: 0 2 6 8; 5: 1 3 7 9; 6: 2 4 8 9; 7: 3 5 8 9; 8: 4 6 7 9; 9:
5 6 7 8}
• Order 12: {0: 1 2 3 4; 1: 0 2 3 4; 2: 0 1 3 5; 3: 0 1 2 5; 4: 0 1 5 6; 5: 2 3 4 7; 6: 4 7 8 9; 7: 5 6 10 11; 8: 6 9 10
11; 9: 6 8 10 11; 10: 7 8 9 11; 11: 7 8 9 10}
• Order 14: {0: 1 2 3 4; 1: 0 2 3 4; 2: 0 1 3 5; 3: 0 1 2 5; 4: 0 1 5 6; 5: 2 3 4 7; 6: 4 8 9 10; 7: 5 8 9 10; 8: 6 7 11
12; 9: 6 7 11 13; 10: 6 7 12 13; 11: 8 9 12 13; 12: 8 10 11 13; 13: 9 10 11 12}
• Order 15: {0: 1 2 3 4; 1: 0 2 3 5; 2: 0 1 6 7; 3: 0 1 6 8; 4: 0 7 9 10; 5: 1 9 11 12; 6: 2 3 10 11; 7: 2 4 13 14; 8: 3
9 13 14; 9: 4 5 8 11; 10: 4 6 12 13; 11: 5 6 9 14; 12: 5 10 13 14; 13: 7 8 10 12; 14: 7 8 11 12}
• Order 17: {0: 1 2 3 4; 1: 0 2 3 4; 2: 0 1 3 5; 3: 0 1 2 6; 4: 0 1 5 7; 5: 2 4 6 8; 6: 3 5 9 10; 7: 4 9 11 12; 8: 5 11
13 14; 9: 6 7 10 13; 10: 6 9 14 15; 11: 7 8 12 16; 12: 7 11 13 16; 13: 8 9 12 15; 14: 8 10 15 16; 15: 10 13 14 16;
16: 11 12 14 15}
• Order 19: {0: 1 2 3 4; 1: 0 2 3 4; 2: 0 1 3 4; 3: 0 1 2 5; 4: 0 1 2 6; 5: 3 6 7 8; 6: 4 5 7 9; 7: 5 6 8 10; 8: 5 7 9 11;
9: 6 8 11 12; 10: 7 11 13 14; 11: 8 9 10 13; 12: 9 13 15 16; 13: 10 11 12 14; 14: 10 13 17 18; 15: 12 16 17 18; 16:
12 15 17 18; 17: 14 15 16 18; 18: 14 15 16 17}
• Order 21: {0: 1 2 3 4; 1: 0 2 3 5; 2: 0 1 6 7; 3: 0 1 8 9; 4: 0 6 10 11; 5: 1 10 12 13; 6: 2 4 14 15; 7: 2 12 14 16; 8:
3 12 16 17; 9: 3 13 14 18; 10: 4 5 16 19; 11: 4 15 18 19; 12: 5 7 8 15; 13: 5 9 18 20; 14: 6 7 9 18; 15: 6 11 12 20;
16: 7 8 10 17; 17: 8 16 19 20; 18: 9 11 13 14; 19: 10 11 17 20; 20: 13 15 17 19}
9
Adjacency lists of 5-regular nut graphs
• Order 10: {0: 1 2 3 4 5; 1: 0 2 3 6 7; 2: 0 1 4 6 8; 3: 0 1 4 7 9; 4: 0 2 3 8 9; 5: 0 6 7 8 9; 6: 1 2 5 7 8; 7: 1 3 5 6
9; 8: 2 4 5 6 9; 9: 3 4 5 7 8}
• Order 12: {0: 1 2 3 4 5; 1: 0 2 3 4 6; 2: 0 1 5 7 8; 3: 0 1 6 9 10; 4: 0 1 7 9 11; 5: 0 2 8 9 10; 6: 1 3 7 8 10; 7: 2 4
6 9 11; 8: 2 5 6 10 11; 9: 3 4 5 7 11; 10: 3 5 6 8 11; 11: 4 7 8 9 10}
• Order 14: {0: 1 2 3 4 5; 1: 0 2 3 4 5; 2: 0 1 3 4 6; 3: 0 1 2 7 8; 4: 0 1 2 9 10; 5: 0 1 6 11 12; 6: 2 5 7 8 9; 7: 3 6 8
9 13; 8: 3 6 7 11 12; 9: 4 6 7 10 13; 10: 4 9 11 12 13; 11: 5 8 10 12 13; 12: 5 8 10 11 13; 13: 7 9 10 11 12}
• Order 16: {0: 1 2 3 4 5; 1: 0 2 3 4 5; 2: 0 1 3 4 5; 3: 0 1 2 4 6; 4: 0 1 2 3 7; 5: 0 1 2 8 9; 6: 3 7 8 10 11; 7: 4 6 12
13 14; 8: 5 6 9 10 12; 9: 5 8 10 12 15; 10: 6 8 9 11 15; 11: 6 10 13 14 15; 12: 7 8 9 13 14; 13: 7 11 12 14 15; 14:
7 11 12 13 15; 15: 9 10 11 13 14}
• Order 18: {0: 1 2 3 4 5; 1: 0 2 3 4 5; 2: 0 1 3 4 5; 3: 0 1 2 4 6; 4: 0 1 2 3 6; 5: 0 1 2 7 8; 6: 3 4 9 10 11; 7: 5 9 10
11 12; 8: 5 9 10 13 14; 9: 6 7 8 10 13; 10: 6 7 8 9 13; 11: 6 7 12 15 16; 12: 7 11 15 16 17; 13: 8 9 10 14 17; 14: 8
13 15 16 17; 15: 11 12 14 16 17; 16: 11 12 14 15 17; 17: 12 13 14 15 16}
Adjacency lists of 6-regular nut graphs
• Order 12: {0: 1 2 3 4 5 6; 1: 0 2 3 4 7 8; 2: 0 1 5 7 9 10; 3: 0 1 5 8 9 11; 4: 0 1 6 7 9 11; 5: 0 2 3 6 10 11; 6: 0 4
5 8 9 10; 7: 1 2 4 8 10 11; 8: 1 3 6 7 9 10; 9: 2 3 4 6 8 11; 10: 2 5 6 7 8 11; 11: 3 4 5 7 9 10}
• Order 13: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 7; 2: 0 1 3 4 8 9; 3: 0 1 2 6 10 11; 4: 0 1 2 8 10 12; 5: 0 1 6 7 11 12; 6: 0 3
5 7 8 9; 7: 1 5 6 9 10 11; 8: 2 4 6 9 10 12; 9: 2 6 7 8 11 12; 10: 3 4 7 8 11 12; 11: 3 5 7 9 10 12; 12: 4 5 8 9 10 11}
• Order 14: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 7; 3: 0 1 2 6 8 9; 4: 0 1 2 7 10 11; 5: 0 1 2 8 9 10; 6: 0 1 3
8 11 12; 7: 2 4 8 10 11 13; 8: 3 5 6 7 12 13; 9: 3 5 10 11 12 13; 10: 4 5 7 9 12 13; 11: 4 6 7 9 12 13; 12: 6 8 9 10
11 13; 13: 7 8 9 10 11 12}
• Order 15: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 7; 3: 0 1 2 4 5 8; 4: 0 1 2 3 7 9; 5: 0 1 2 3 10 11; 6: 0 1 10
12 13 14; 7: 2 4 9 12 13 14; 8: 3 9 10 11 12 13; 9: 4 7 8 11 12 14; 10: 5 6 8 11 12 13; 11: 5 8 9 10 13 14; 12: 6 7
8 9 10 14; 13: 6 7 8 10 11 14; 14: 6 7 9 11 12 13}
• Order 16: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 6; 3: 0 1 2 4 5 7; 4: 0 1 2 3 7 8; 5: 0 1 2 3 9 10; 6: 0 1 2 7 9
11; 7: 3 4 6 8 12 13; 8: 4 7 11 12 14 15; 9: 5 6 10 11 14 15; 10: 5 9 11 12 13 14; 11: 6 8 9 10 13 15; 12: 7 8 10 13
14 15; 13: 7 10 11 12 14 15; 14: 8 9 10 12 13 15; 15: 8 9 11 12 13 14}
• Order 17: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 6; 3: 0 1 2 4 5 6; 4: 0 1 2 3 7 8; 5: 0 1 2 3 7 9; 6: 0 1 2 3 10
11; 7: 4 5 10 12 13 14; 8: 4 9 11 12 15 16; 9: 5 8 13 14 15 16; 10: 6 7 11 13 15 16; 11: 6 8 10 14 15 16; 12: 7 8 13
14 15 16; 13: 7 9 10 12 14 15; 14: 7 9 11 12 13 16; 15: 8 9 10 11 12 13; 16: 8 9 10 11 12 14}
• Order 18: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 6; 3: 0 1 2 4 5 6; 4: 0 1 2 3 5 7; 5: 0 1 2 3 4 7; 6: 0 1 2 3 8 9;
7: 4 5 8 9 10 11; 8: 6 7 10 12 13 14; 9: 6 7 12 13 15 16; 10: 7 8 11 14 15 17; 11: 7 10 12 13 14 16; 12: 8 9 11 14 16
17; 13: 8 9 11 15 16 17; 14: 8 10 11 12 15 17; 15: 9 10 13 14 16 17; 16: 9 11 12 13 15 17; 17: 10 12 13 14 15 16}
• Order 19: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 6; 3: 0 1 2 4 5 6; 4: 0 1 2 3 5 6; 5: 0 1 2 3 4 7; 6: 0 1 2 3 4
8; 7: 5 8 9 10 11 12; 8: 6 7 9 10 13 14; 9: 7 8 10 11 12 15; 10: 7 8 9 13 16 17; 11: 7 9 12 14 15 16; 12: 7 9 11 15
17 18; 13: 8 10 14 16 17 18; 14: 8 11 13 15 16 18; 15: 9 11 12 14 17 18; 16: 10 11 13 14 17 18; 17: 10 12 13 15 16
18; 18: 12 13 14 15 16 17}
• Order 20: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 6; 3: 0 1 2 4 7 8; 4: 0 1 2 3 7 9; 5: 0 1 2 6 7 8; 6: 0 1 2 5 7
9; 7: 3 4 5 6 8 10; 8: 3 5 7 9 10 11; 9: 4 6 8 10 11 12; 10: 7 8 9 12 13 14; 11: 8 9 12 13 14 15; 12: 9 10 11 16 17
18; 13: 10 11 15 16 17 19; 14: 10 11 15 16 18 19; 15: 11 13 14 17 18 19; 16: 12 13 14 17 18 19; 17: 12 13 15 16
18 19; 18: 12 14 15 16 17 19; 19: 13 14 15 16 17 18}
• Order 21: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 6; 3: 0 1 2 4 7 8; 4: 0 1 2 3 7 9; 5: 0 1 2 6 7 8; 6: 0 1 2 5 7
9; 7: 3 4 5 6 8 9; 8: 3 5 7 9 10 11; 9: 4 6 7 8 10 11; 10: 8 9 12 13 14 15; 11: 8 9 16 17 18 19; 12: 10 13 14 15 16
17; 13: 10 12 14 16 18 20; 14: 10 12 13 18 19 20; 15: 10 12 16 17 19 20; 16: 11 12 13 15 17 19; 17: 11 12 15 16
18 20; 18: 11 13 14 17 19 20; 19: 11 14 15 16 18 20; 20: 13 14 15 17 18 19}
• Order 22: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 6; 3: 0 1 2 4 7 8; 4: 0 1 2 3 7 9; 5: 0 1 2 6 7 8; 6: 0 1 2 5 7
9; 7: 3 4 5 6 8 9; 8: 3 5 7 9 10 11; 9: 4 6 7 8 10 11; 10: 8 9 11 12 13 14; 11: 8 9 10 12 15 16; 12: 10 11 13 17 18
19; 13: 10 12 15 16 20 21; 14: 10 15 17 18 20 21; 15: 11 13 14 16 17 20; 16: 11 13 15 17 19 21; 17: 12 14 15 16
18 19; 18: 12 14 17 19 20 21; 19: 12 16 17 18 20 21; 20: 13 14 15 18 19 21; 21: 13 14 16 18 19 20}
• Order 23: {0: 1 2 3 4 5 6; 1: 0 2 3 4 5 6; 2: 0 1 3 4 5 6; 3: 0 1 2 4 7 8; 4: 0 1 2 3 7 9; 5: 0 1 2 6 7 8; 6: 0 1 2 5 7
9; 7: 3 4 5 6 8 9; 8: 3 5 7 9 10 11; 9: 4 6 7 8 10 11; 10: 8 9 11 12 13 14; 11: 8 9 10 12 13 15; 12: 10 11 13 14 16
17; 13: 10 11 12 15 16 18; 14: 10 12 15 17 18 19; 15: 11 13 14 17 20 21; 16: 12 13 17 20 21 22; 17: 12 14 15 16
19 22; 18: 13 14 19 20 21 22; 19: 14 17 18 20 21 22; 20: 15 16 18 19 21 22; 21: 15 16 18 19 20 22; 22: 16 17 18
19 20 21}
10
Adjacency lists of 7-regular nut graphs
• Order 12: {0: 1 2 3 4 5 6 7; 1: 0 2 3 4 5 8 9; 2: 0 1 3 4 6 10 11; 3: 0 1 2 5 8 10 11; 4: 0 1 2 7 8 9 10; 5: 0 1 3 6 7
9 11; 6: 0 2 5 7 9 10 11; 7: 0 4 5 6 8 9 10; 8: 1 3 4 7 9 10 11; 9: 1 4 5 6 7 8 11; 10: 2 3 4 6 7 8 11; 11: 2 3 5 6 8 9
10}
• Order 14: {0: 1 2 3 4 5 6 7; 1: 0 2 3 4 5 6 7; 2: 0 1 3 4 5 8 9; 3: 0 1 2 4 5 8 10; 4: 0 1 2 3 6 9 11; 5: 0 1 2 3 11 12
13; 6: 0 1 4 9 10 12 13; 7: 0 1 8 10 11 12 13; 8: 2 3 7 9 10 11 12; 9: 2 4 6 8 10 12 13; 10: 3 6 7 8 9 11 13; 11: 4 5
7 8 10 12 13; 12: 5 6 7 8 9 11 13; 13: 5 6 7 9 10 11 12}
• Order 16: {0: 1 2 3 4 5 6 7; 1: 0 2 3 4 5 6 7; 2: 0 1 3 4 5 6 7; 3: 0 1 2 4 5 6 8; 4: 0 1 2 3 7 9 10; 5: 0 1 2 3 8 9
11; 6: 0 1 2 3 8 12 13; 7: 0 1 2 4 10 11 12; 8: 3 5 6 11 13 14 15; 9: 4 5 11 12 13 14 15; 10: 4 7 11 12 13 14 15; 11:
5 7 8 9 10 14 15; 12: 6 7 9 10 13 14 15; 13: 6 8 9 10 12 14 15; 14: 8 9 10 11 12 13 15; 15: 8 9 10 11 12 13 14}
• Order 18: {0: 1 2 3 4 5 6 7; 1: 0 2 3 4 5 6 7; 2: 0 1 3 4 5 6 7; 3: 0 1 2 4 5 6 7; 4: 0 1 2 3 5 6 8; 5: 0 1 2 3 4 7 9;
6: 0 1 2 3 4 8 10; 7: 0 1 2 3 5 9 11; 8: 4 6 10 11 12 13 14; 9: 5 7 11 12 13 15 16; 10: 6 8 11 14 15 16 17; 11: 7 8 9
10 12 15 17; 12: 8 9 11 13 14 16 17; 13: 8 9 12 14 15 16 17; 14: 8 10 12 13 15 16 17; 15: 9 10 11 13 14 16 17; 16:
9 10 12 13 14 15 17; 17: 10 11 12 13 14 15 16}
• Order 20: {0: 1 2 3 4 5 6 7; 1: 0 2 3 4 5 6 7; 2: 0 1 3 4 5 6 7; 3: 0 1 2 4 5 6 7; 4: 0 1 2 3 5 6 7; 5: 0 1 2 3 4 6 8;
6: 0 1 2 3 4 5 8; 7: 0 1 2 3 4 8 9; 8: 5 6 7 10 11 12 13; 9: 7 10 14 15 16 17 18; 10: 8 9 11 12 14 15 19; 11: 8 10 12
13 14 15 19; 12: 8 10 11 16 17 18 19; 13: 8 11 14 15 16 17 18; 14: 9 10 11 13 15 16 17; 15: 9 10 11 13 14 18 19;
16: 9 12 13 14 17 18 19; 17: 9 12 13 14 16 18 19; 18: 9 12 13 15 16 17 19; 19: 10 11 12 15 16 17 18}
• Order 22: {0: 1 2 3 4 5 6 7; 1: 0 2 3 4 5 6 7; 2: 0 1 3 4 5 6 8; 3: 0 1 2 4 7 8 9; 4: 0 1 2 3 10 11 12; 5: 0 1 2 6 7 8
9; 6: 0 1 2 5 10 11 12; 7: 0 1 3 5 8 9 10; 8: 2 3 5 7 9 10 11; 9: 3 5 7 8 10 12 13; 10: 4 6 7 8 9 13 14; 11: 4 6 8 15
16 17 18; 12: 4 6 9 13 14 15 19; 13: 9 10 12 16 19 20 21; 14: 10 12 15 16 17 18 20; 15: 11 12 14 16 17 19 21; 16:
11 13 14 15 18 20 21; 17: 11 14 15 18 19 20 21; 18: 11 14 16 17 19 20 21; 19: 12 13 15 17 18 20 21; 20: 13 14 16
17 18 19 21; 21: 13 15 16 17 18 19 20}
• Order 24: {0: 1 2 3 4 5 8 9; 1: 0 2 3 4 5 10 11; 2: 0 1 5 6 7 12 13; 3: 0 1 4 6 7 14 15; 4: 0 1 3 6 7 16 17; 5: 0 1 2
6 7 18 19; 6: 2 3 4 5 7 22 23; 7: 2 3 4 5 6 20 21; 8: 0 9 16 18 20 22 23; 9: 0 8 17 19 21 22 23; 10: 1 11 12 14 20
21 22; 11: 1 10 13 15 20 21 23; 12: 2 10 13 14 16 17 22; 13: 2 11 12 15 16 17 23; 14: 3 10 12 15 18 19 22; 15: 3
11 13 14 18 19 23; 16: 4 8 12 13 17 18 20; 17: 4 9 12 13 16 19 21; 18: 5 8 14 15 16 19 20; 19: 5 9 14 15 17 18 21;
20: 7 8 10 11 16 18 21; 21: 7 9 10 11 17 19 20; 22: 6 8 9 10 12 14 23; 23: 6 8 9 11 13 15 22}
Adjacency lists of 8-regular nut graphs
• Order 12: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 9; 2: 0 1 3 4 5 8 9 10; 3: 0 1 2 4 6 8 10 11; 4: 0 1 2 3 7 9 10 11; 5:
0 1 2 6 7 8 10 11; 6: 0 1 3 5 7 9 10 11; 7: 0 1 4 5 6 8 9 11; 8: 0 2 3 5 7 9 10 11; 9: 1 2 4 6 7 8 10 11; 10: 2 3 4 5 6
8 9 11; 11: 3 4 5 6 7 8 9 10}
• Order 14: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 9 10; 3: 0 1 2 4 5 11 12 13; 4: 0 1 2 3 7 8 9 11; 5:
0 1 2 3 9 10 11 12; 6: 0 1 2 7 10 11 12 13; 7: 0 1 4 6 9 10 12 13; 8: 0 1 4 9 10 11 12 13; 9: 2 4 5 7 8 10 11 13; 10:
2 5 6 7 8 9 12 13; 11: 3 4 5 6 8 9 12 13; 12: 3 5 6 7 8 10 11 13; 13: 3 6 7 8 9 10 11 12}
• Order 15: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 9; 3: 0 1 2 4 5 6 8 10; 4: 0 1 2 3 5 9 11 12; 5: 0 1
2 3 4 9 11 13; 6: 0 1 2 3 10 11 12 14; 7: 0 1 2 10 11 12 13 14; 8: 0 1 3 9 10 12 13 14; 9: 2 4 5 8 10 12 13 14; 10: 3
6 7 8 9 11 13 14; 11: 4 5 6 7 10 12 13 14; 12: 4 6 7 8 9 11 13 14; 13: 5 7 8 9 10 11 12 14; 14: 6 7 8 9 10 11 12 13}
• Order 16: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 9; 4: 0 1 2 3 5 9 10 11; 5: 0 1
2 3 4 12 13 14; 6: 0 1 2 3 10 12 13 15; 7: 0 1 2 3 10 12 14 15; 8: 0 1 2 9 11 13 14 15; 9: 3 4 8 10 11 12 13 15; 10:
4 6 7 9 11 12 13 14; 11: 4 8 9 10 12 13 14 15; 12: 5 6 7 9 10 11 14 15; 13: 5 6 8 9 10 11 14 15; 14: 5 7 8 10 11 12
13 15; 15: 6 7 8 9 11 12 13 14}
• Order 17: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 9 10; 5: 0 1
2 3 4 9 11 12; 6: 0 1 2 3 4 10 13 14; 7: 0 1 2 3 9 13 15 16; 8: 0 1 2 3 11 12 13 15; 9: 4 5 7 10 11 13 14 16; 10: 4 6
9 11 12 14 15 16; 11: 5 8 9 10 12 14 15 16; 12: 5 8 10 11 13 14 15 16; 13: 6 7 8 9 12 14 15 16; 14: 6 9 10 11 12
13 15 16; 15: 7 8 10 11 12 13 14 16; 16: 7 9 10 11 12 13 14 15}
• Order 18: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 9; 5: 0 1 2
3 4 6 9 10; 6: 0 1 2 3 4 5 11 12; 7: 0 1 2 3 4 9 13 14; 8: 0 1 2 3 9 13 15 16; 9: 4 5 7 8 10 11 15 17; 10: 5 9 11 12
14 15 16 17; 11: 6 9 10 12 13 14 16 17; 12: 6 10 11 13 14 15 16 17; 13: 7 8 11 12 14 15 16 17; 14: 7 10 11 12 13
15 16 17; 15: 8 9 10 12 13 14 16 17; 16: 8 10 11 12 13 14 15 17; 17: 9 10 11 12 13 14 15 16}
• Order 19: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 8; 5: 0 1 2
3 4 9 10 11; 6: 0 1 2 3 4 9 12 13; 7: 0 1 2 3 4 10 14 15; 8: 0 1 2 3 4 12 16 17; 9: 5 6 10 11 14 15 16 17; 10: 5 7 9
11 12 14 15 18; 11: 5 9 10 12 13 16 17 18; 12: 6 8 10 11 13 14 16 18; 13: 6 11 12 14 15 16 17 18; 14: 7 9 10 12 13
15 17 18; 15: 7 9 10 13 14 16 17 18; 16: 8 9 11 12 13 15 17 18; 17: 8 9 11 13 14 15 16 18; 18: 10 11 12 13 14 15
16 17}
• Order 20: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 8; 5: 0 1 2
3 4 6 7 8; 6: 0 1 2 3 4 5 9 10; 7: 0 1 2 3 4 5 9 11; 8: 0 1 2 3 4 5 12 13; 9: 6 7 10 11 12 14 15 16; 10: 6 9 12 13 14
15 17 18; 11: 7 9 13 14 15 16 17 19; 12: 8 9 10 14 16 17 18 19; 13: 8 10 11 15 16 17 18 19; 14: 9 10 11 12 16 17
18 19; 15: 9 10 11 13 16 17 18 19; 16: 9 11 12 13 14 15 18 19; 17: 10 11 12 13 14 15 18 19; 18: 10 12 13 14 15 16
17 19; 19: 11 12 13 14 15 16 17 18}
11
• Order 21: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 8; 5: 0 1 2
3 4 6 9 10; 6: 0 1 2 3 4 5 9 11; 7: 0 1 2 3 4 8 9 10; 8: 0 1 2 3 4 7 9 11; 9: 5 6 7 8 12 13 14 15; 10: 5 7 12 13 16 17
18 19; 11: 6 8 12 14 16 17 18 20; 12: 9 10 11 15 16 17 19 20; 13: 9 10 14 15 16 18 19 20; 14: 9 11 13 15 16 18 19
20; 15: 9 12 13 14 17 18 19 20; 16: 10 11 12 13 14 17 19 20; 17: 10 11 12 15 16 18 19 20; 18: 10 11 13 14 15 17
19 20; 19: 10 12 13 14 15 16 17 18; 20: 11 12 13 14 15 16 17 18}
• Order 22: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 8; 5: 0 1 2
3 4 6 9 10; 6: 0 1 2 3 4 5 9 11; 7: 0 1 2 3 4 8 9 10; 8: 0 1 2 3 4 7 9 11; 9: 5 6 7 8 10 11 12 13; 10: 5 7 9 11 12 13
14 15; 11: 6 8 9 10 12 14 16 17; 12: 9 10 11 13 15 18 19 20; 13: 9 10 12 14 16 18 19 21; 14: 10 11 13 15 17 18 20
21; 15: 10 12 14 16 17 19 20 21; 16: 11 13 15 17 18 19 20 21; 17: 11 14 15 16 18 19 20 21; 18: 12 13 14 16 17 19
20 21; 19: 12 13 15 16 17 18 20 21; 20: 12 14 15 16 17 18 19 21; 21: 13 14 15 16 17 18 19 20}
• Order 23: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 8; 5: 0 1 2
3 4 6 9 10; 6: 0 1 2 3 4 5 9 11; 7: 0 1 2 3 4 8 9 10; 8: 0 1 2 3 4 7 9 11; 9: 5 6 7 8 10 11 12 13; 10: 5 7 9 11 12 13
14 15; 11: 6 8 9 10 12 13 14 15; 12: 9 10 11 13 14 15 16 17; 13: 9 10 11 12 16 18 19 20; 14: 10 11 12 15 17 18 21
22; 15: 10 11 12 14 19 20 21 22; 16: 12 13 17 18 19 20 21 22; 17: 12 14 16 18 19 20 21 22; 18: 13 14 16 17 19 20
21 22; 19: 13 15 16 17 18 20 21 22; 20: 13 15 16 17 18 19 21 22; 21: 14 15 16 17 18 19 20 22; 22: 14 15 16 17 18
19 20 21}
• Order 24: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 8; 5: 0 1 2
3 4 6 9 10; 6: 0 1 2 3 4 5 9 11; 7: 0 1 2 3 4 8 9 10; 8: 0 1 2 3 4 7 9 11; 9: 5 6 7 8 10 11 12 13; 10: 5 7 9 11 12 13
14 15; 11: 6 8 9 10 12 13 14 15; 12: 9 10 11 13 14 15 16 17; 13: 9 10 11 12 14 16 17 18; 14: 10 11 12 13 15 19 20
21; 15: 10 11 12 14 16 19 22 23; 16: 12 13 15 18 20 21 22 23; 17: 12 13 18 19 20 21 22 23; 18: 13 16 17 19 20 21
22 23; 19: 14 15 17 18 20 21 22 23; 20: 14 16 17 18 19 21 22 23; 21: 14 16 17 18 19 20 22 23; 22: 15 16 17 18 19
20 21 23; 23: 15 16 17 18 19 20 21 22}
• Order 25: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 8; 5: 0 1 2
3 4 6 9 10; 6: 0 1 2 3 4 5 9 11; 7: 0 1 2 3 4 8 9 10; 8: 0 1 2 3 4 7 9 11; 9: 5 6 7 8 10 11 12 13; 10: 5 7 9 11 12 13
14 15; 11: 6 8 9 10 12 13 14 15; 12: 9 10 11 13 14 15 16 17; 13: 9 10 11 12 14 15 18 19; 14: 10 11 12 13 16 17 20
21; 15: 10 11 12 13 16 20 22 23; 16: 12 14 15 18 19 21 22 24; 17: 12 14 18 20 21 22 23 24; 18: 13 16 17 19 21 22
23 24; 19: 13 16 18 20 21 22 23 24; 20: 14 15 17 19 21 22 23 24; 21: 14 16 17 18 19 20 23 24; 22: 15 16 17 18 19
20 23 24; 23: 15 17 18 19 20 21 22 24; 24: 16 17 18 19 20 21 22 23}
• Order 26: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 9 10; 4: 0 1 2 3 5 9 11 12; 5: 0
1 2 3 4 11 12 13; 6: 0 1 2 3 11 14 15 16; 7: 0 1 2 8 9 10 11 12; 8: 0 1 2 7 9 10 11 12; 9: 3 4 7 8 10 11 12 13; 10: 3
7 8 9 11 12 13 14; 11: 4 5 6 7 8 9 10 12; 12: 4 5 7 8 9 10 11 13; 13: 5 9 10 12 14 15 16 17; 14: 6 10 13 15 16 17
18 19; 15: 6 13 14 16 17 18 19 20; 16: 6 13 14 15 18 21 22 23; 17: 13 14 15 19 21 22 24 25; 18: 14 15 16 20 21 23
24 25; 19: 14 15 17 20 22 23 24 25; 20: 15 18 19 21 22 23 24 25; 21: 16 17 18 20 22 23 24 25; 22: 16 17 19 20 21
23 24 25; 23: 16 18 19 20 21 22 24 25; 24: 17 18 19 20 21 22 23 25; 25: 17 18 19 20 21 22 23 24}
• Order 27: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 7 8; 5: 0 1 2
3 4 6 9 10; 6: 0 1 2 3 4 5 9 11; 7: 0 1 2 3 4 8 9 10; 8: 0 1 2 3 4 7 9 11; 9: 5 6 7 8 10 11 12 13; 10: 5 7 9 11 12 13
14 15; 11: 6 8 9 10 12 13 14 15; 12: 9 10 11 13 14 15 16 17; 13: 9 10 11 12 14 15 16 17; 14: 10 11 12 13 15 16 17
18; 15: 10 11 12 13 14 16 18 19; 16: 12 13 14 15 17 20 21 22; 17: 12 13 14 16 18 19 20 23; 18: 14 15 17 19 23 24
25 26; 19: 15 17 18 21 22 24 25 26; 20: 16 17 21 22 23 24 25 26; 21: 16 19 20 22 23 24 25 26; 22: 16 19 20 21 23
24 25 26; 23: 17 18 20 21 22 24 25 26; 24: 18 19 20 21 22 23 25 26; 25: 18 19 20 21 22 23 24 26; 26: 18 19 20 21
22 23 24 25}
• Order 29: {0: 1 2 3 4 5 6 7 8; 1: 0 2 3 4 5 6 7 8; 2: 0 1 3 4 5 6 7 8; 3: 0 1 2 4 5 6 7 8; 4: 0 1 2 3 5 6 9 10; 5: 0 1
2 3 4 7 9 10; 6: 0 1 2 3 4 9 10 11; 7: 0 1 2 3 5 9 12 13; 8: 0 1 2 3 9 11 12 14; 9: 4 5 6 7 8 10 11 12; 10: 4 5 6 9 11
12 13 14; 11: 6 8 9 10 12 13 14 15; 12: 7 8 9 10 11 13 14 15; 13: 7 10 11 12 14 15 16 17; 14: 8 10 11 12 13 15 16
17; 15: 11 12 13 14 16 17 18 19; 16: 13 14 15 17 18 19 20 21; 17: 13 14 15 16 18 19 20 22; 18: 15 16 17 21 22 23
24 25; 19: 15 16 17 23 24 26 27 28; 20: 16 17 21 23 25 26 27 28; 21: 16 18 20 22 23 24 25 26; 22: 17 18 21 23 24
25 27 28; 23: 18 19 20 21 22 26 27 28; 24: 18 19 21 22 25 26 27 28; 25: 18 20 21 22 24 26 27 28; 26: 19 20 21 23
24 25 27 28; 27: 19 20 22 23 24 25 26 28; 28: 19 20 22 23 24 25 26 27}
Adjacency lists of 9-regular nut graphs
• Order 16: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 7 10 11; 4: 0 1 2 3 5 8 10
12 13; 5: 0 1 2 3 4 11 12 13 14; 6: 0 1 2 3 9 11 12 14 15; 7: 0 1 2 3 10 11 12 13 15; 8: 0 1 2 4 9 10 13 14 15; 9: 0
1 2 6 8 11 13 14 15; 10: 3 4 7 8 11 12 13 14 15; 11: 3 5 6 7 9 10 12 14 15; 12: 4 5 6 7 10 11 13 14 15; 13: 4 5 7 8
9 10 12 14 15; 14: 5 6 8 9 10 11 12 13 15; 15: 6 7 8 9 10 11 12 13 14}
• Order 18: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 7 8 9; 4: 0 1 2 3 5 6 7 8
10; 5: 0 1 2 3 4 6 11 12 13; 6: 0 1 2 3 4 5 14 15 16; 7: 0 1 2 3 4 10 11 14 17; 8: 0 1 2 3 4 11 12 15 16; 9: 0 1 2 3
10 11 12 13 17; 10: 4 7 9 12 13 14 15 16 17; 11: 5 7 8 9 13 14 15 16 17; 12: 5 8 9 10 13 14 15 16 17; 13: 5 9 10
11 12 14 15 16 17; 14: 6 7 10 11 12 13 15 16 17; 15: 6 8 10 11 12 13 14 16 17; 16: 6 8 10 11 12 13 14 15 17; 17: 7
9 10 11 12 13 14 15 16}
12
• Order 20: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 7 8 9; 4: 0 1 2 3 5 6 7 8 9;
5: 0 1 2 3 4 6 7 8 10; 6: 0 1 2 3 4 5 10 11 12; 7: 0 1 2 3 4 5 11 13 14; 8: 0 1 2 3 4 5 12 13 15; 9: 0 1 2 3 4 11 14
15 16; 10: 5 6 12 14 15 16 17 18 19; 11: 6 7 9 12 13 15 17 18 19; 12: 6 8 10 11 13 16 17 18 19; 13: 7 8 11 12 14
16 17 18 19; 14: 7 9 10 13 15 16 17 18 19; 15: 8 9 10 11 14 16 17 18 19; 16: 9 10 12 13 14 15 17 18 19; 17: 10 11
12 13 14 15 16 18 19; 18: 10 11 12 13 14 15 16 17 19; 19: 10 11 12 13 14 15 16 17 18}
• Order 22: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 7 8 9; 4: 0 1 2 3 5 6 7 8 9;
5: 0 1 2 3 4 6 7 8 9; 6: 0 1 2 3 4 5 7 8 10; 7: 0 1 2 3 4 5 6 8 10; 8: 0 1 2 3 4 5 6 7 11; 9: 0 1 2 3 4 5 11 12 13; 10:
6 7 12 13 14 15 16 17 18; 11: 8 9 12 14 15 16 19 20 21; 12: 9 10 11 13 14 15 17 19 20; 13: 9 10 12 14 16 17 18 19
21; 14: 10 11 12 13 15 16 18 20 21; 15: 10 11 12 14 17 18 19 20 21; 16: 10 11 13 14 17 18 19 20 21; 17: 10 12 13
15 16 18 19 20 21; 18: 10 13 14 15 16 17 19 20 21; 19: 11 12 13 15 16 17 18 20 21; 20: 11 12 14 15 16 17 18 19
21; 21: 11 13 14 15 16 17 18 19 20}
• Order 24: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 10 11 12; 4: 0 1 2 3 5 6
10 11 13; 5: 0 1 2 3 4 14 15 16 17; 6: 0 1 2 3 4 18 19 20 21; 7: 0 1 2 8 10 14 15 18 19; 8: 0 1 2 7 10 14 15 18 19;
9: 0 1 2 10 11 12 13 14 15; 10: 3 4 7 8 9 11 12 13 14; 11: 3 4 9 10 12 13 14 15 16; 12: 3 9 10 11 13 14 15 16 22;
13: 4 9 10 11 12 17 20 22 23; 14: 5 7 8 9 10 11 12 21 22; 15: 5 7 8 9 11 12 18 20 23; 16: 5 11 12 17 19 20 21 22
23; 17: 5 13 16 18 19 20 21 22 23; 18: 6 7 8 15 17 20 21 22 23; 19: 6 7 8 16 17 20 21 22 23; 20: 6 13 15 16 17 18
19 21 23; 21: 6 14 16 17 18 19 20 22 23; 22: 12 13 14 16 17 18 19 21 23; 23: 13 15 16 17 18 19 20 21 22}
• Order 26: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 7 8 9; 4: 0 1 2 3 5 6 7 8
10; 5: 0 1 2 3 4 6 7 11 12; 6: 0 1 2 3 4 5 11 13 14; 7: 0 1 2 3 4 5 15 16 17; 8: 0 1 2 3 4 9 13 15 16; 9: 0 1 2 3 8 10
11 12 13; 10: 4 9 11 12 13 14 15 16 17; 11: 5 6 9 10 12 13 14 15 16; 12: 5 9 10 11 13 14 15 16 17; 13: 6 8 9 10 11
12 18 19 20; 14: 6 10 11 12 15 18 19 21 22; 15: 7 8 10 11 12 14 18 21 23; 16: 7 8 10 11 12 17 20 24 25; 17: 7 10
12 16 20 22 23 24 25; 18: 13 14 15 19 21 22 23 24 25; 19: 13 14 18 20 21 22 23 24 25; 20: 13 16 17 19 21 22 23
24 25; 21: 14 15 18 19 20 22 23 24 25; 22: 14 17 18 19 20 21 23 24 25; 23: 15 17 18 19 20 21 22 24 25; 24: 16 17
18 19 20 21 22 23 25; 25: 16 17 18 19 20 21 22 23 24}
• Order 28: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 10 11 12; 4: 0 1 2 3 5 6 10
11 13; 5: 0 1 2 3 4 10 12 13 14; 6: 0 1 2 3 4 12 15 16 17; 7: 0 1 2 8 9 10 12 15 18; 8: 0 1 2 7 9 10 12 15 18; 9: 0 1
2 7 8 10 12 15 18; 10: 3 4 5 7 8 9 11 12 13; 11: 3 4 10 12 13 14 15 16 17; 12: 3 5 6 7 8 9 10 11 13; 13: 4 5 10 11
12 14 15 16 17; 14: 5 11 13 15 16 17 18 19 20; 15: 6 7 8 9 11 13 14 16 19; 16: 6 11 13 14 15 19 21 22 23; 17: 6 11
13 14 19 20 24 25 26; 18: 7 8 9 14 21 22 23 24 27; 19: 14 15 16 17 20 21 25 26 27; 20: 14 17 19 22 23 24 25 26 27;
21: 16 18 19 22 23 24 25 26 27; 22: 16 18 20 21 23 24 25 26 27; 23: 16 18 20 21 22 24 25 26 27; 24: 17 18 20 21
22 23 25 26 27; 25: 17 19 20 21 22 23 24 26 27; 26: 17 19 20 21 22 23 24 25 27; 27: 18 19 20 21 22 23 24 25 26}
• Order 30: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 7 8 9; 4: 0 1 2 3 5 6 7 8 9;
5: 0 1 2 3 4 6 7 8 9; 6: 0 1 2 3 4 5 7 10 11; 7: 0 1 2 3 4 5 6 12 13; 8: 0 1 2 3 4 5 10 11 12; 9: 0 1 2 3 4 5 10 14 15;
10: 6 8 9 12 13 16 17 18 19; 11: 6 8 13 20 21 22 23 24 25; 12: 7 8 10 13 14 15 16 17 18; 13: 7 10 11 12 14 15 16
17 18; 14: 9 12 13 15 16 17 18 19 20; 15: 9 12 13 14 16 17 18 19 20; 16: 10 12 13 14 15 17 18 19 20; 17: 10 12 13
14 15 16 18 19 20; 18: 10 12 13 14 15 16 17 21 26; 19: 10 14 15 16 17 22 23 24 26; 20: 11 14 15 16 17 22 27 28
29; 21: 11 18 23 24 25 26 27 28 29; 22: 11 19 20 23 25 26 27 28 29; 23: 11 19 21 22 24 25 27 28 29; 24: 11 19 21
23 25 26 27 28 29; 25: 11 21 22 23 24 26 27 28 29; 26: 18 19 21 22 24 25 27 28 29; 27: 20 21 22 23 24 25 26 28
29; 28: 20 21 22 23 24 25 26 27 29; 29: 20 21 22 23 24 25 26 27 28}
• Order 32: {0: 1 2 3 4 5 6 7 8 9; 1: 0 2 3 4 5 6 7 8 9; 2: 0 1 3 4 5 6 7 8 9; 3: 0 1 2 4 5 6 7 8 9; 4: 0 1 2 3 5 6 7 8 9;
5: 0 1 2 3 4 6 7 8 9; 6: 0 1 2 3 4 5 7 10 11; 7: 0 1 2 3 4 5 6 12 13; 8: 0 1 2 3 4 5 10 11 12; 9: 0 1 2 3 4 5 10 14 15;
10: 6 8 9 11 14 16 17 18 19; 11: 6 8 10 12 13 15 16 20 21; 12: 7 8 11 13 14 15 16 17 18; 13: 7 11 12 14 15 16 17
18 19; 14: 9 10 12 13 15 16 17 18 19; 15: 9 11 12 13 14 16 17 18 19; 16: 10 11 12 13 14 15 17 18 19; 17: 10 12 13
14 15 16 18 19 20; 18: 10 12 13 14 15 16 17 19 22; 19: 10 13 14 15 16 17 18 23 24; 20: 11 17 21 22 23 25 26 27
28; 21: 11 20 22 23 25 26 27 29 30; 22: 18 20 21 23 24 25 29 30 31; 23: 19 20 21 22 24 25 28 29 31; 24: 19 22 23
26 27 28 29 30 31; 25: 20 21 22 23 26 27 28 30 31; 26: 20 21 24 25 27 28 29 30 31; 27: 20 21 24 25 26 28 29 30
31; 28: 20 23 24 25 26 27 29 30 31; 29: 21 22 23 24 26 27 28 30 31; 30: 21 22 24 25 26 27 28 29 31; 31: 22 23 24
25 26 27 28 29 30}
Adjacency lists of 10-regular nut graphs
• Order 15: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 11; 2: 0 1 3 4 5 6 7 8 10 12; 3: 0 1 2 4 5 6 7 9 11 13; 4: 0
1 2 3 5 6 7 10 13 14; 5: 0 1 2 3 4 6 9 11 12 14; 6: 0 1 2 3 4 5 11 12 13 14; 7: 0 1 2 3 4 8 10 12 13 14; 8: 0 1 2 7 9
10 11 12 13 14; 9: 0 1 3 5 8 10 11 12 13 14; 10: 0 2 4 7 8 9 11 12 13 14; 11: 1 3 5 6 8 9 10 12 13 14; 12: 2 5 6 7 8
9 10 11 13 14; 13: 3 4 6 7 8 9 10 11 12 14; 14: 4 5 6 7 8 9 10 11 12 13}
• Order 16: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 11; 3: 0 1 2 4 5 6 7 8 10 11; 4: 0 1
2 3 5 6 11 12 13 14; 5: 0 1 2 3 4 7 12 13 14 15; 6: 0 1 2 3 4 8 12 13 14 15; 7: 0 1 2 3 5 9 10 11 12 15; 8: 0 1 2 3 6
9 12 13 14 15; 9: 0 1 2 7 8 10 11 13 14 15; 10: 0 1 3 7 9 11 12 13 14 15; 11: 2 3 4 7 9 10 12 13 14 15; 12: 4 5 6 7
8 10 11 13 14 15; 13: 4 5 6 8 9 10 11 12 14 15; 14: 4 5 6 8 9 10 11 12 13 15; 15: 5 6 7 8 9 10 11 12 13 14}
• Order 17: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 11; 4: 0 1
2 3 5 6 7 10 12 13; 5: 0 1 2 3 4 8 12 13 14 15; 6: 0 1 2 3 4 10 11 12 14 16; 7: 0 1 2 3 4 12 13 14 15 16; 8: 0 1 2 3
5 11 13 14 15 16; 9: 0 1 2 3 11 12 13 14 15 16; 10: 0 1 2 4 6 11 12 14 15 16; 11: 3 6 8 9 10 12 13 14 15 16; 12: 4
5 6 7 9 10 11 13 15 16; 13: 4 5 7 8 9 11 12 14 15 16; 14: 5 6 7 8 9 10 11 13 15 16; 15: 5 7 8 9 10 11 12 13 14 16;
16: 6 7 8 9 10 11 12 13 14 15}
13
• Order 18: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1
2 3 5 6 7 8 11 12; 5: 0 1 2 3 4 6 9 13 14 15; 6: 0 1 2 3 4 5 10 13 16 17; 7: 0 1 2 3 4 11 12 13 14 15; 8: 0 1 2 3 4
13 14 15 16 17; 9: 0 1 2 3 5 11 12 14 16 17; 10: 0 1 2 3 6 11 12 15 16 17; 11: 4 7 9 10 12 13 14 15 16 17; 12: 4 7
9 10 11 13 14 15 16 17; 13: 5 6 7 8 11 12 14 15 16 17; 14: 5 7 8 9 11 12 13 15 16 17; 15: 5 7 8 10 11 12 13 14 16
17; 16: 6 8 9 10 11 12 13 14 15 17; 17: 6 8 9 10 11 12 13 14 15 16}
• Order 19: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1
2 3 5 6 7 8 9 11; 5: 0 1 2 3 4 6 7 10 11 12; 6: 0 1 2 3 4 5 10 13 14 15; 7: 0 1 2 3 4 5 12 13 16 17; 8: 0 1 2 3 4 11
14 16 17 18; 9: 0 1 2 3 4 13 15 16 17 18; 10: 0 1 2 3 5 6 12 14 15 18; 11: 4 5 8 12 13 14 15 16 17 18; 12: 5 7 10
11 13 14 15 16 17 18; 13: 6 7 9 11 12 14 15 16 17 18; 14: 6 8 10 11 12 13 15 16 17 18; 15: 6 9 10 11 12 13 14 16
17 18; 16: 7 8 9 11 12 13 14 15 17 18; 17: 7 8 9 11 12 13 14 15 16 18; 18: 8 9 10 11 12 13 14 15 16 17}
• Order 20: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1
2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 11 12; 6: 0 1 2 3 4 5 7 9 11 13; 7: 0 1 2 3 4 5 6 14 15 16; 8: 0 1 2 3 4 5 11 13
17 18; 9: 0 1 2 3 4 6 12 14 15 19; 10: 0 1 2 3 4 13 16 17 18 19; 11: 5 6 8 12 14 15 16 17 18 19; 12: 5 9 11 13 14
15 16 17 18 19; 13: 6 8 10 12 14 15 16 17 18 19; 14: 7 9 11 12 13 15 16 17 18 19; 15: 7 9 11 12 13 14 16 17 18
19; 16: 7 10 11 12 13 14 15 17 18 19; 17: 8 10 11 12 13 14 15 16 18 19; 18: 8 10 11 12 13 14 15 16 17 19; 19: 9
10 11 12 13 14 15 16 17 18}
• Order 21: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1
2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 11; 6: 0 1 2 3 4 5 7 8 10 12; 7: 0 1 2 3 4 5 6 11 13 14; 8: 0 1 2 3 4 5 6 15 16
17; 9: 0 1 2 3 4 5 13 14 18 19; 10: 0 1 2 3 4 6 12 13 15 20; 11: 5 7 12 14 15 16 17 18 19 20; 12: 6 10 11 13 15 16
17 18 19 20; 13: 7 9 10 12 14 16 17 18 19 20; 14: 7 9 11 13 15 16 17 18 19 20; 15: 8 10 11 12 14 16 17 18 19 20;
16: 8 11 12 13 14 15 17 18 19 20; 17: 8 11 12 13 14 15 16 18 19 20; 18: 9 11 12 13 14 15 16 17 19 20; 19: 9 11 12
13 14 15 16 17 18 20; 20: 10 11 12 13 14 15 16 17 18 19}
• Order 22: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1
2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 10; 6: 0 1 2 3 4 5 7 8 9 11; 7: 0 1 2 3 4 5 6 8 11 12; 8: 0 1 2 3 4 5 6 7 11 13;
9: 0 1 2 3 4 5 6 14 15 16; 10: 0 1 2 3 4 5 11 12 14 17; 11: 6 7 8 10 13 15 18 19 20 21; 12: 7 10 13 14 16 17 18 19
20 21; 13: 8 11 12 15 16 17 18 19 20 21; 14: 9 10 12 15 16 17 18 19 20 21; 15: 9 11 13 14 16 17 18 19 20 21; 16:
9 12 13 14 15 17 18 19 20 21; 17: 10 12 13 14 15 16 18 19 20 21; 18: 11 12 13 14 15 16 17 19 20 21; 19: 11 12 13
14 15 16 17 18 20 21; 20: 11 12 13 14 15 16 17 18 19 21; 21: 11 12 13 14 15 16 17 18 19 20}
• Order 23: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1
2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 11 12 13; 6: 0 1 2 3 4 5 8 11 12 14; 7: 0 1 2 3 4 5 11 15 16 17; 8: 0 1 2 3 4 6 13
18 19 20; 9: 0 1 2 3 4 10 11 12 13 14; 10: 0 1 2 3 4 9 11 12 13 15; 11: 5 6 7 9 10 15 16 18 19 21; 12: 5 6 9 10 13
16 17 18 19 22; 13: 5 8 9 10 12 14 18 20 21 22; 14: 6 9 13 15 16 17 19 20 21 22; 15: 7 10 11 14 17 18 19 20 21
22; 16: 7 11 12 14 17 18 19 20 21 22; 17: 7 12 14 15 16 18 19 20 21 22; 18: 8 11 12 13 15 16 17 20 21 22; 19: 8
11 12 14 15 16 17 20 21 22; 20: 8 13 14 15 16 17 18 19 21 22; 21: 11 13 14 15 16 17 18 19 20 22; 22: 12 13 14 15
16 17 18 19 20 21}
• Order 24: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1
2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 11 12 13; 6: 0 1 2 3 4 5 8 11 12 14; 7: 0 1 2 3 4 5 11 15 16 17; 8: 0 1 2 3 4 6 12
18 19 20; 9: 0 1 2 3 4 10 11 12 13 14; 10: 0 1 2 3 4 9 11 12 13 14; 11: 5 6 7 9 10 12 13 14 15 18; 12: 5 6 8 9 10
11 15 16 19 21; 13: 5 9 10 11 17 18 20 21 22 23; 14: 6 9 10 11 15 17 19 20 22 23; 15: 7 11 12 14 16 19 20 21 22
23; 16: 7 12 15 17 18 19 20 21 22 23; 17: 7 13 14 16 18 19 20 21 22 23; 18: 8 11 13 16 17 19 20 21 22 23; 19: 8
12 14 15 16 17 18 21 22 23; 20: 8 13 14 15 16 17 18 21 22 23; 21: 12 13 15 16 17 18 19 20 22 23; 22: 13 14 15 16
17 18 19 20 21 23; 23: 13 14 15 16 17 18 19 20 21 22}
• Order 25: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1
2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 11 12 13; 6: 0 1 2 3 4 5 8 11 12 14; 7: 0 1 2 3 4 5 11 15 16 17; 8: 0 1 2 3 4 6 12
15 16 18; 9: 0 1 2 3 4 10 11 12 13 14; 10: 0 1 2 3 4 9 11 12 13 14; 11: 5 6 7 9 10 12 13 14 15 16; 12: 5 6 8 9 10
11 13 15 17 19; 13: 5 9 10 11 12 17 19 20 21 22; 14: 6 9 10 11 19 20 21 22 23 24; 15: 7 8 11 12 17 18 20 21 23
24; 16: 7 8 11 17 18 19 20 22 23 24; 17: 7 12 13 15 16 18 21 22 23 24; 18: 8 15 16 17 19 20 21 22 23 24; 19: 12
13 14 16 18 20 21 22 23 24; 20: 13 14 15 16 18 19 21 22 23 24; 21: 13 14 15 17 18 19 20 22 23 24; 22: 13 14 16
17 18 19 20 21 23 24; 23: 14 15 16 17 18 19 20 21 22 24; 24: 14 15 16 17 18 19 20 21 22 23}
• Order 26: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1 2
3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 11 12 13; 6: 0 1 2 3 4 5 8 11 12 14; 7: 0 1 2 3 4 5 11 15 16 17; 8: 0 1 2 3 4 6 12 15
16 18; 9: 0 1 2 3 4 10 11 12 13 14; 10: 0 1 2 3 4 9 11 12 13 14; 11: 5 6 7 9 10 12 13 14 15 16; 12: 5 6 8 9 10 11 13
14 15 18; 13: 5 9 10 11 12 15 16 19 20 21; 14: 6 9 10 11 12 17 19 20 21 22; 15: 7 8 11 12 13 16 22 23 24 25; 16: 7
8 11 13 15 17 19 23 24 25; 17: 7 14 16 18 20 21 22 23 24 25; 18: 8 12 17 19 20 21 22 23 24 25; 19: 13 14 16 18 20
21 22 23 24 25; 20: 13 14 17 18 19 21 22 23 24 25; 21: 13 14 17 18 19 20 22 23 24 25; 22: 14 15 17 18 19 20 21 23
24 25; 23: 15 16 17 18 19 20 21 22 24 25; 24: 15 16 17 18 19 20 21 22 23 25; 25: 15 16 17 18 19 20 21 22 23 24}
• Order 27: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1
2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 11 12 13; 6: 0 1 2 3 4 5 8 11 12 14; 7: 0 1 2 3 4 5 11 15 16 17; 8: 0 1 2 3 4 6 12
15 16 18; 9: 0 1 2 3 4 10 11 12 13 14; 10: 0 1 2 3 4 9 11 12 13 14; 11: 5 6 7 9 10 12 13 14 15 16; 12: 5 6 8 9 10
11 13 14 15 16; 13: 5 9 10 11 12 14 15 16 19 20; 14: 6 9 10 11 12 13 15 21 22 23; 15: 7 8 11 12 13 14 19 20 21
24; 16: 7 8 11 12 13 17 18 24 25 26; 17: 7 16 18 19 20 21 22 23 25 26; 18: 8 16 17 19 21 22 23 24 25 26; 19: 13
15 17 18 20 22 23 24 25 26; 20: 13 15 17 19 21 22 23 24 25 26; 21: 14 15 17 18 20 22 23 24 25 26; 22: 14 17 18
19 20 21 23 24 25 26; 23: 14 17 18 19 20 21 22 24 25 26; 24: 15 16 18 19 20 21 22 23 25 26; 25: 16 17 18 19 20
21 22 23 24 26; 26: 16 17 18 19 20 21 22 23 24 25}
14
• Order 28: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1
2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 10; 6: 0 1 2 3 4 5 7 8 11 12; 7: 0 1 2 3 4 5 6 11 12 13; 8: 0 1 2 3 4 5 6 11 13
14; 9: 0 1 2 3 4 5 11 14 15 16; 10: 0 1 2 3 4 5 12 14 17 18; 11: 6 7 8 9 12 13 14 15 16 17; 12: 6 7 10 11 13 14 15
16 17 18; 13: 7 8 11 12 14 15 16 17 18 19; 14: 8 9 10 11 12 13 15 16 17 19; 15: 9 11 12 13 14 17 18 20 21 22; 16:
9 11 12 13 14 20 21 23 24 25; 17: 10 11 12 13 14 15 22 23 26 27; 18: 10 12 13 15 20 22 24 25 26 27; 19: 13 14 20
21 22 23 24 25 26 27; 20: 15 16 18 19 21 23 24 25 26 27; 21: 15 16 19 20 22 23 24 25 26 27; 22: 15 17 18 19 21
23 24 25 26 27; 23: 16 17 19 20 21 22 24 25 26 27; 24: 16 18 19 20 21 22 23 25 26 27; 25: 16 18 19 20 21 22 23
24 26 27; 26: 17 18 19 20 21 22 23 24 25 27; 27: 17 18 19 20 21 22 23 24 25 26}
• Order 29: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1
2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 10; 6: 0 1 2 3 4 5 7 8 11 12; 7: 0 1 2 3 4 5 6 11 12 13; 8: 0 1 2 3 4 5 6 11 13
14; 9: 0 1 2 3 4 5 11 14 15 16; 10: 0 1 2 3 4 5 12 14 17 18; 11: 6 7 8 9 12 13 14 15 16 17; 12: 6 7 10 11 13 14 15
16 17 18; 13: 7 8 11 12 14 15 16 17 18 19; 14: 8 9 10 11 12 13 15 16 17 18; 15: 9 11 12 13 14 16 17 19 20 21; 16:
9 11 12 13 14 15 22 23 24 25; 17: 10 11 12 13 14 15 20 22 26 27; 18: 10 12 13 14 20 23 24 26 27 28; 19: 13 15 20
21 22 23 25 26 27 28; 20: 15 17 18 19 21 22 23 24 25 28; 21: 15 19 20 22 23 24 25 26 27 28; 22: 16 17 19 20 21 24
25 26 27 28; 23: 16 18 19 20 21 24 25 26 27 28; 24: 16 18 20 21 22 23 25 26 27 28; 25: 16 19 20 21 22 23 24 26
27 28; 26: 17 18 19 21 22 23 24 25 27 28; 27: 17 18 19 21 22 23 24 25 26 28; 28: 18 19 20 21 22 23 24 25 26 27}
• Order 30: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1
2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 10; 6: 0 1 2 3 4 5 7 8 11 12; 7: 0 1 2 3 4 5 6 11 12 13; 8: 0 1 2 3 4 5 6 11 13
14; 9: 0 1 2 3 4 5 11 14 15 16; 10: 0 1 2 3 4 5 12 14 17 18; 11: 6 7 8 9 12 13 14 15 16 17; 12: 6 7 10 11 13 14 15
16 17 18; 13: 7 8 11 12 14 15 16 17 18 19; 14: 8 9 10 11 12 13 15 16 17 18; 15: 9 11 12 13 14 16 17 18 19 20; 16:
9 11 12 13 14 15 17 18 19 21; 17: 10 11 12 13 14 15 16 19 20 22; 18: 10 12 13 14 15 16 21 22 23 24; 19: 13 15 16
17 23 25 26 27 28 29; 20: 15 17 21 23 24 25 26 27 28 29; 21: 16 18 20 22 24 25 26 27 28 29; 22: 17 18 21 23 24
25 26 27 28 29; 23: 18 19 20 22 24 25 26 27 28 29; 24: 18 20 21 22 23 25 26 27 28 29; 25: 19 20 21 22 23 24 26
27 28 29; 26: 19 20 21 22 23 24 25 27 28 29; 27: 19 20 21 22 23 24 25 26 28 29; 28: 19 20 21 22 23 24 25 26 27
29; 29: 19 20 21 22 23 24 25 26 27 28}
• Order 31: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1
2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 10; 6: 0 1 2 3 4 5 7 8 11 12; 7: 0 1 2 3 4 5 6 11 12 13; 8: 0 1 2 3 4 5 6 11 13
14; 9: 0 1 2 3 4 5 11 14 15 16; 10: 0 1 2 3 4 5 12 14 17 18; 11: 6 7 8 9 12 13 14 15 16 17; 12: 6 7 10 11 13 14 15
16 17 18; 13: 7 8 11 12 14 15 16 17 18 19; 14: 8 9 10 11 12 13 15 16 17 18; 15: 9 11 12 13 14 16 17 18 19 20; 16:
9 11 12 13 14 15 17 18 19 20; 17: 10 11 12 13 14 15 16 19 21 22; 18: 10 12 13 14 15 16 21 23 24 25; 19: 13 15 16
17 20 23 26 27 28 29; 20: 15 16 19 22 23 24 25 26 27 30; 21: 17 18 22 23 24 25 26 28 29 30; 22: 17 20 21 23 24
26 27 28 29 30; 23: 18 19 20 21 22 25 27 28 29 30; 24: 18 20 21 22 25 26 27 28 29 30; 25: 18 20 21 23 24 26 27
28 29 30; 26: 19 20 21 22 24 25 27 28 29 30; 27: 19 20 22 23 24 25 26 28 29 30; 28: 19 21 22 23 24 25 26 27 29
30; 29: 19 21 22 23 24 25 26 27 28 30; 30: 20 21 22 23 24 25 26 27 28 29}
• Order 32: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1
2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 10; 6: 0 1 2 3 4 5 7 8 11 12; 7: 0 1 2 3 4 5 6 11 12 13; 8: 0 1 2 3 4 5 6 11 13
14; 9: 0 1 2 3 4 5 11 14 15 16; 10: 0 1 2 3 4 5 12 14 17 18; 11: 6 7 8 9 12 13 14 15 16 17; 12: 6 7 10 11 13 14 15
16 17 18; 13: 7 8 11 12 14 15 16 17 18 19; 14: 8 9 10 11 12 13 15 16 17 18; 15: 9 11 12 13 14 16 17 18 19 20; 16:
9 11 12 13 14 15 17 18 19 20; 17: 10 11 12 13 14 15 16 18 19 20; 18: 10 12 13 14 15 16 17 19 21 22; 19: 13 15 16
17 18 20 21 23 24 25; 20: 15 16 17 19 21 22 23 26 27 28; 21: 18 19 20 23 26 27 28 29 30 31; 22: 18 20 23 24 25
26 27 29 30 31; 23: 19 20 21 22 24 25 28 29 30 31; 24: 19 22 23 25 26 27 28 29 30 31; 25: 19 22 23 24 26 27 28
29 30 31; 26: 20 21 22 24 25 27 28 29 30 31; 27: 20 21 22 24 25 26 28 29 30 31; 28: 20 21 23 24 25 26 27 29 30
31; 29: 21 22 23 24 25 26 27 28 30 31; 30: 21 22 23 24 25 26 27 28 29 31; 31: 21 22 23 24 25 26 27 28 29 30}
• Order 33: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1
2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 10; 6: 0 1 2 3 4 5 7 8 11 12; 7: 0 1 2 3 4 5 6 11 12 13; 8: 0 1 2 3 4 5 6 11 13
14; 9: 0 1 2 3 4 5 11 14 15 16; 10: 0 1 2 3 4 5 12 14 17 18; 11: 6 7 8 9 12 13 14 15 16 17; 12: 6 7 10 11 13 14 15
16 17 18; 13: 7 8 11 12 14 15 16 17 18 19; 14: 8 9 10 11 12 13 15 16 17 18; 15: 9 11 12 13 14 16 17 18 19 20; 16:
9 11 12 13 14 15 17 18 19 20; 17: 10 11 12 13 14 15 16 18 19 20; 18: 10 12 13 14 15 16 17 19 21 22; 19: 13 15 16
17 18 23 24 25 26 27; 20: 15 16 17 21 23 24 28 29 30 31; 21: 18 20 23 25 26 28 29 30 31 32; 22: 18 23 24 25 26
27 28 29 30 32; 23: 19 20 21 22 24 25 27 28 29 32; 24: 19 20 22 23 26 27 28 29 30 31; 25: 19 21 22 23 26 27 28
30 31 32; 26: 19 21 22 24 25 27 29 30 31 32; 27: 19 22 23 24 25 26 29 30 31 32; 28: 20 21 22 23 24 25 29 30 31
32; 29: 20 21 22 23 24 26 27 28 31 32; 30: 20 21 22 24 25 26 27 28 31 32; 31: 20 21 24 25 26 27 28 29 30 32; 32:
21 22 23 25 26 27 28 29 30 31}
• Order 34: {0: 1 2 3 4 5 6 7 8 9 10; 1: 0 2 3 4 5 6 7 8 9 10; 2: 0 1 3 4 5 6 7 8 9 10; 3: 0 1 2 4 5 6 7 8 9 10; 4: 0 1
2 3 5 6 7 8 9 10; 5: 0 1 2 3 4 6 7 8 9 10; 6: 0 1 2 3 4 5 7 8 11 12; 7: 0 1 2 3 4 5 6 11 12 13; 8: 0 1 2 3 4 5 6 11 13
14; 9: 0 1 2 3 4 5 11 14 15 16; 10: 0 1 2 3 4 5 12 14 17 18; 11: 6 7 8 9 12 13 14 15 16 17; 12: 6 7 10 11 13 14 15
16 17 18; 13: 7 8 11 12 14 15 16 17 18 19; 14: 8 9 10 11 12 13 15 16 17 18; 15: 9 11 12 13 14 16 17 18 19 20; 16:
9 11 12 13 14 15 17 18 19 20; 17: 10 11 12 13 14 15 16 18 19 20; 18: 10 12 13 14 15 16 17 19 20 21; 19: 13 15 16
17 18 20 21 22 23 24; 20: 15 16 17 18 19 21 22 25 26 27; 21: 18 19 20 22 23 24 25 28 29 30; 22: 19 20 21 23 25
26 27 28 31 32; 23: 19 21 22 25 26 27 29 31 32 33; 24: 19 21 25 26 28 29 30 31 32 33; 25: 20 21 22 23 24 26 28
29 30 33; 26: 20 22 23 24 25 27 30 31 32 33; 27: 20 22 23 26 28 29 30 31 32 33; 28: 21 22 24 25 27 29 30 31 32
33; 29: 21 23 24 25 27 28 30 31 32 33; 30: 21 24 25 26 27 28 29 31 32 33; 31: 22 23 24 26 27 28 29 30 32 33; 32:
22 23 24 26 27 28 29 30 31 33; 33: 23 24 25 26 27 28 29 30 31 32}
15
Adjacency lists of 11-regular nut graphs
• Order 16: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 12; 3: 0 1 2 4 5 6 7 11 12
13 14; 4: 0 1 2 3 5 6 8 11 13 14 15; 5: 0 1 2 3 4 9 10 11 12 13 15; 6: 0 1 2 3 4 9 11 12 13 14 15; 7: 0 1 2 3 8 9 10
11 12 14 15; 8: 0 1 2 4 7 9 10 12 13 14 15; 9: 0 1 2 5 6 7 8 12 13 14 15; 10: 0 1 2 5 7 8 11 12 13 14 15; 11: 0 1 3
4 5 6 7 10 13 14 15; 12: 2 3 5 6 7 8 9 10 13 14 15; 13: 3 4 5 6 8 9 10 11 12 14 15; 14: 3 4 6 7 8 9 10 11 12 13 15;
15: 4 5 6 7 8 9 10 11 12 13 14}
• Order 18: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9
10 11; 4: 0 1 2 3 5 6 7 8 12 13 14; 5: 0 1 2 3 4 6 7 9 12 13 15; 6: 0 1 2 3 4 5 10 12 14 16 17; 7: 0 1 2 3 4 5 11 14
15 16 17; 8: 0 1 2 3 4 12 13 14 15 16 17; 9: 0 1 2 3 5 12 13 14 15 16 17; 10: 0 1 2 3 6 11 12 13 15 16 17; 11: 0 1
2 3 7 10 13 14 15 16 17; 12: 4 5 6 8 9 10 13 14 15 16 17; 13: 4 5 8 9 10 11 12 14 15 16 17; 14: 4 6 7 8 9 11 12 13
15 16 17; 15: 5 7 8 9 10 11 12 13 14 16 17; 16: 6 7 8 9 10 11 12 13 14 15 17; 17: 6 7 8 9 10 11 12 13 14 15 16}
• Order 20: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9
10 11; 4: 0 1 2 3 5 6 7 8 9 10 11; 5: 0 1 2 3 4 6 7 8 12 13 14; 6: 0 1 2 3 4 5 12 13 15 16 17; 7: 0 1 2 3 4 5 12 14
15 16 18; 8: 0 1 2 3 4 5 12 15 17 18 19; 9: 0 1 2 3 4 12 13 14 17 18 19; 10: 0 1 2 3 4 13 14 15 16 17 19; 11: 0 1 2
3 4 13 14 15 16 18 19; 12: 5 6 7 8 9 13 15 16 17 18 19; 13: 5 6 9 10 11 12 14 16 17 18 19; 14: 5 7 9 10 11 13 15
16 17 18 19; 15: 6 7 8 10 11 12 14 16 17 18 19; 16: 6 7 10 11 12 13 14 15 17 18 19; 17: 6 8 9 10 12 13 14 15 16
18 19; 18: 7 8 9 11 12 13 14 15 16 17 19; 19: 8 9 10 11 12 13 14 15 16 17 18}
• Order 22: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9
10 11; 4: 0 1 2 3 5 6 7 8 9 10 12; 5: 0 1 2 3 4 6 7 8 9 12 13; 6: 0 1 2 3 4 5 11 12 14 15 16; 7: 0 1 2 3 4 5 11 13 14
15 17; 8: 0 1 2 3 4 5 12 13 14 17 18; 9: 0 1 2 3 4 5 12 13 17 19 20; 10: 0 1 2 3 4 11 14 17 18 19 21; 11: 0 1 2 3 6
7 10 15 16 20 21; 12: 4 5 6 8 9 15 16 18 19 20 21; 13: 5 7 8 9 14 15 16 18 19 20 21; 14: 6 7 8 10 13 16 17 18 19
20 21; 15: 6 7 11 12 13 16 17 18 19 20 21; 16: 6 11 12 13 14 15 17 18 19 20 21; 17: 7 8 9 10 14 15 16 18 19 20
21; 18: 8 10 12 13 14 15 16 17 19 20 21; 19: 9 10 12 13 14 15 16 17 18 20 21; 20: 9 11 12 13 14 15 16 17 18 19
21; 21: 10 11 12 13 14 15 16 17 18 19 20}
• Order 24: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9 10
11; 4: 0 1 2 3 5 6 7 8 9 10 11; 5: 0 1 2 3 4 6 7 8 9 10 12; 6: 0 1 2 3 4 5 7 8 9 13 14; 7: 0 1 2 3 4 5 6 10 15 16 17;
8: 0 1 2 3 4 5 6 10 15 16 18; 9: 0 1 2 3 4 5 6 10 15 19 20; 10: 0 1 2 3 4 5 7 8 9 13 21; 11: 0 1 2 3 4 13 14 17 18 22
23; 12: 5 13 14 15 16 17 19 20 21 22 23; 13: 6 10 11 12 17 18 19 20 21 22 23; 14: 6 11 12 15 16 17 18 19 20 21 22;
15: 7 8 9 12 14 16 17 19 20 22 23; 16: 7 8 12 14 15 17 18 19 21 22 23; 17: 7 11 12 13 14 15 16 18 20 21 23; 18: 8
11 13 14 16 17 19 20 21 22 23; 19: 9 12 13 14 15 16 18 20 21 22 23; 20: 9 12 13 14 15 17 18 19 21 22 23; 21: 10
12 13 14 16 17 18 19 20 22 23; 22: 11 12 13 14 15 16 18 19 20 21 23; 23: 11 12 13 15 16 17 18 19 20 21 22}
• Order 26: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9
10 11; 4: 0 1 2 3 5 6 7 8 9 10 11; 5: 0 1 2 3 4 6 7 8 9 12 13; 6: 0 1 2 3 4 5 10 11 14 15 16; 7: 0 1 2 3 4 5 10 14 15
16 17; 8: 0 1 2 3 4 5 10 14 15 18 19; 9: 0 1 2 3 4 5 18 19 20 21 22; 10: 0 1 2 3 4 6 7 8 12 13 14; 11: 0 1 2 3 4 6 12
13 14 15 16; 12: 5 10 11 13 14 15 16 17 18 19 20; 13: 5 10 11 12 14 15 18 21 22 23 24; 14: 6 7 8 10 11 12 13 19
20 23 25; 15: 6 7 8 11 12 13 17 21 23 24 25; 16: 6 7 11 12 17 19 20 22 23 24 25; 17: 7 12 15 16 18 20 21 22 23 24
25; 18: 8 9 12 13 17 19 20 21 22 24 25; 19: 8 9 12 14 16 18 21 22 23 24 25; 20: 9 12 14 16 17 18 21 22 23 24 25;
21: 9 13 15 17 18 19 20 22 23 24 25; 22: 9 13 16 17 18 19 20 21 23 24 25; 23: 13 14 15 16 17 19 20 21 22 24 25;
24: 13 15 16 17 18 19 20 21 22 23 25; 25: 14 15 16 17 18 19 20 21 22 23 24}
• Order 28: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9
10 11; 4: 0 1 2 3 5 6 7 8 9 10 11; 5: 0 1 2 3 4 6 7 8 9 12 13; 6: 0 1 2 3 4 5 10 11 14 15 16; 7: 0 1 2 3 4 5 10 12 17
18 19; 8: 0 1 2 3 4 5 10 14 15 20 21; 9: 0 1 2 3 4 5 11 14 15 16 20; 10: 0 1 2 3 4 6 7 8 12 13 14; 11: 0 1 2 3 4 6 9
12 13 14 15; 12: 5 7 10 11 13 14 15 16 17 18 19; 13: 5 10 11 12 14 15 16 17 18 19 20; 14: 6 8 9 10 11 12 13 16 17
22 23; 15: 6 8 9 11 12 13 18 22 24 25 26; 16: 6 9 12 13 14 19 22 23 24 25 27; 17: 7 12 13 14 21 22 23 24 25 26
27; 18: 7 12 13 15 20 21 22 24 25 26 27; 19: 7 12 13 16 21 22 23 24 25 26 27; 20: 8 9 13 18 21 22 23 24 25 26 27;
21: 8 17 18 19 20 22 23 24 25 26 27; 22: 14 15 16 17 18 19 20 21 23 26 27; 23: 14 16 17 19 20 21 22 24 25 26 27;
24: 15 16 17 18 19 20 21 23 25 26 27; 25: 15 16 17 18 19 20 21 23 24 26 27; 26: 15 17 18 19 20 21 22 23 24 25
27; 27: 16 17 18 19 20 21 22 23 24 25 26}
• Order 30: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9 10
11; 4: 0 1 2 3 5 6 7 8 9 10 11; 5: 0 1 2 3 4 6 7 8 9 12 13; 6: 0 1 2 3 4 5 10 11 14 15 16; 7: 0 1 2 3 4 5 10 12 17 18
19; 8: 0 1 2 3 4 5 10 14 15 16 20; 9: 0 1 2 3 4 5 13 21 22 23 24; 10: 0 1 2 3 4 6 7 8 12 13 14; 11: 0 1 2 3 4 6 12 13
14 15 16; 12: 5 7 10 11 13 14 15 16 17 18 19; 13: 5 9 10 11 12 14 15 16 17 18 19; 14: 6 8 10 11 12 13 15 16 17 18
19; 15: 6 8 11 12 13 14 16 17 18 19 20; 16: 6 8 11 12 13 14 15 17 21 22 25; 17: 7 12 13 14 15 16 18 20 23 25 26;
18: 7 12 13 14 15 17 21 25 27 28 29; 19: 7 12 13 14 15 23 24 26 27 28 29; 20: 8 15 17 22 23 24 25 26 27 28 29; 21:
9 16 18 22 23 24 25 26 27 28 29; 22: 9 16 20 21 23 24 25 26 27 28 29; 23: 9 17 19 20 21 22 24 26 27 28 29; 24: 9
19 20 21 22 23 25 26 27 28 29; 25: 16 17 18 20 21 22 24 26 27 28 29; 26: 17 19 20 21 22 23 24 25 27 28 29; 27:
18 19 20 21 22 23 24 25 26 28 29; 28: 18 19 20 21 22 23 24 25 26 27 29; 29: 18 19 20 21 22 23 24 25 26 27 28}
• Order 32: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9
10 11; 4: 0 1 2 3 5 6 7 8 9 10 11; 5: 0 1 2 3 4 6 7 8 9 12 13; 6: 0 1 2 3 4 5 10 11 14 15 16; 7: 0 1 2 3 4 5 10 12 17
18 19; 8: 0 1 2 3 4 5 10 14 15 16 20; 9: 0 1 2 3 4 5 12 17 18 21 22; 10: 0 1 2 3 4 6 7 8 12 13 14; 11: 0 1 2 3 4 6 12
13 14 15 16; 12: 5 7 9 10 11 13 14 15 16 17 18; 13: 5 10 11 12 14 15 16 17 18 19 20; 14: 6 8 10 11 12 13 15 16 17
18 19; 15: 6 8 11 12 13 14 16 17 18 19 20; 16: 6 8 11 12 13 14 15 17 18 19 20; 17: 7 9 12 13 14 15 16 18 19 20
23; 18: 7 9 12 13 14 15 16 17 21 24 25; 19: 7 13 14 15 16 17 23 24 25 26 27; 20: 8 13 15 16 17 22 23 28 29 30 31;
16
21: 9 18 23 24 25 26 27 28 29 30 31; 22: 9 20 23 24 25 26 27 28 29 30 31; 23: 17 19 20 21 22 26 27 28 29 30 31;
24: 18 19 21 22 25 26 27 28 29 30 31; 25: 18 19 21 22 24 26 27 28 29 30 31; 26: 19 21 22 23 24 25 27 28 29 30
31; 27: 19 21 22 23 24 25 26 28 29 30 31; 28: 20 21 22 23 24 25 26 27 29 30 31; 29: 20 21 22 23 24 25 26 27 28
30 31; 30: 20 21 22 23 24 25 26 27 28 29 31; 31: 20 21 22 23 24 25 26 27 28 29 30}
• Order 34: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9
10 11; 4: 0 1 2 3 5 6 7 8 9 10 11; 5: 0 1 2 3 4 6 7 8 9 12 13; 6: 0 1 2 3 4 5 10 11 14 15 16; 7: 0 1 2 3 4 5 10 12 17
18 19; 8: 0 1 2 3 4 5 10 14 15 16 20; 9: 0 1 2 3 4 5 12 17 18 20 21; 10: 0 1 2 3 4 6 7 8 12 13 14; 11: 0 1 2 3 4 6 12
13 14 15 16; 12: 5 7 9 10 11 13 14 15 16 17 18; 13: 5 10 11 12 14 15 16 17 18 19 20; 14: 6 8 10 11 12 13 15 16 17
18 19; 15: 6 8 11 12 13 14 16 17 18 19 20; 16: 6 8 11 12 13 14 15 17 18 19 20; 17: 7 9 12 13 14 15 16 18 19 20
21; 18: 7 9 12 13 14 15 16 17 19 20 22; 19: 7 13 14 15 16 17 18 20 21 23 24; 20: 8 9 13 15 16 17 18 19 25 26 27;
21: 9 17 19 22 23 24 25 28 29 30 31; 22: 18 21 23 24 25 26 27 28 29 32 33; 23: 19 21 22 25 26 27 28 30 31 32 33;
24: 19 21 22 25 26 28 29 30 31 32 33; 25: 20 21 22 23 24 27 29 30 31 32 33; 26: 20 22 23 24 27 28 29 30 31 32
33; 27: 20 22 23 25 26 28 29 30 31 32 33; 28: 21 22 23 24 26 27 29 30 31 32 33; 29: 21 22 24 25 26 27 28 30 31
32 33; 30: 21 23 24 25 26 27 28 29 31 32 33; 31: 21 23 24 25 26 27 28 29 30 32 33; 32: 22 23 24 25 26 27 28 29
30 31 33; 33: 22 23 24 25 26 27 28 29 30 31 32}
• Order 36: {0: 1 2 3 4 5 6 7 8 9 10 11; 1: 0 2 3 4 5 6 7 8 9 10 11; 2: 0 1 3 4 5 6 7 8 9 10 11; 3: 0 1 2 4 5 6 7 8 9 10
11; 4: 0 1 2 3 5 6 7 8 9 10 11; 5: 0 1 2 3 4 6 7 8 9 12 13; 6: 0 1 2 3 4 5 10 11 14 15 16; 7: 0 1 2 3 4 5 10 12 17 18
19; 8: 0 1 2 3 4 5 10 14 15 16 20; 9: 0 1 2 3 4 5 12 17 18 20 21; 10: 0 1 2 3 4 6 7 8 12 13 14; 11: 0 1 2 3 4 6 12 13
14 15 16; 12: 5 7 9 10 11 13 14 15 16 17 18; 13: 5 10 11 12 14 15 16 17 18 19 20; 14: 6 8 10 11 12 13 15 16 17 18
19; 15: 6 8 11 12 13 14 16 17 18 19 20; 16: 6 8 11 12 13 14 15 17 18 19 20; 17: 7 9 12 13 14 15 16 18 19 20 21;
18: 7 9 12 13 14 15 16 17 19 20 21; 19: 7 13 14 15 16 17 18 20 21 22 23; 20: 8 9 13 15 16 17 18 19 21 22 24; 21:
9 17 18 19 20 22 23 24 25 26 27; 22: 19 20 21 23 24 25 26 27 28 29 30; 23: 19 21 22 25 28 29 31 32 33 34 35; 24:
20 21 22 26 28 29 31 32 33 34 35; 25: 21 22 23 27 28 29 30 31 32 33 34; 26: 21 22 24 27 28 29 30 31 32 33 35; 27:
21 22 25 26 28 30 31 32 33 34 35; 28: 22 23 24 25 26 27 29 30 31 34 35; 29: 22 23 24 25 26 28 30 32 33 34 35; 30:
22 25 26 27 28 29 31 32 33 34 35; 31: 23 24 25 26 27 28 30 32 33 34 35; 32: 23 24 25 26 27 29 30 31 33 34 35; 33:
23 24 25 26 27 29 30 31 32 34 35; 34: 23 24 25 27 28 29 30 31 32 33 35; 35: 23 24 26 27 28 29 30 31 32 33 34}
17