On 12-regular nut graphs

On 12-regular nut graphs arXiv:2102.04418v1 [math.CO] 8 Feb 2021 Nino Bašić∗ Martin Knor† Riste Škrekovski‡ February 9, 2021 Abstract A nut graph is a simple graph whose adjacency matrix is singular with 1-dimensional kernel such that the corresponding eigenvector has no zero entries. In 2020, Fowler et al. characterised for each d ∈ {3, 4, . . . , 11} all values n such that there exists a d-regular nut graph of order n. In the present paper, we determine all values n for which a 12-regular nut graph of order n exists. We also present a result by which there are infinitely many circulant nut graphs of degree d ≡ 0 (mod 4) and no circulant nut graph of degree d ≡ 2 (mod 4). Keywords: Nut graph, adjacency matrix, singular matrix, core graph, Fowler construction, regular graph. Math. Subj. Class. (2020): 05C50, 15A18. 1 Introduction Let G be a simple graph with the vertex set V (G) = {0, 1, . . . , n−1}. Its adjacency matrix A is a symmetric n × n matrix with entries ai,j , where 0 ≤ i, j ≤ n−1, such that ai,j = aj,i = 1 if {i, j} is an edge of G and ai,j = aj,i = 0 otherwise. Graph G is a nut graph if A has eigenvalue 0, the eigenspace corresponding to the eigenvalue 0 is 1-dimensional and generated by an eigenvector which does not contain a 0 entry. Observe that if the eigenspace corresponding to 0 is more than 1-dimensional, then there exists an eigenvector containing entry 0 that is different from 0 = (0, 0, . . . , 0)T . For an introductory treatment of spectral graph theory, which links graphs to linear algebra, see e.g. [3, 6, 7]. Nut graphs have been studied in [4, 8, 11, 12, 14, 15, 16, 17, 18, 20], see also the webpage https://hog.grinvin.org/Nuts within the House of Graphs [2, 5]. Recently, this concept was ∗ FAMNIT & IAM, University of Primorska, 6000 Koper, Slovenia, and Institute of Mathematics, Physics and Mechanics, 1000 Ljubljana, Slovenia, email: nino.basic@famnit.upr.si. † Department of Mathematics, Faculty of Civil Engineering, Slovak University of Technology in Bratislava, Radlinského 11, 810 05, Bratislava, Slovakia, email: knor@math.sk. ‡ Faculty of Information Studies, 8000 Novo mesto, Slovenia, and FMF, University of Ljubljana, 1000 Ljubljana, Slovenia, and FAMNIT, University of Primorska, 6000 Koper, Slovenia, email: skrekovski@gmail.com. 1 extended to signed graphs [1]. Nut graphs have chemical applications, see e.g. [8, 10, 19]. However, in the present paper we consider 12-regular graphs, so our motivation is purely mathematical. In [20], Gutman and Sciriha showed that the smallest non-trivial nut graph has order 7. In [9], Fowler et al. determined all nut graphs on up to 10 vertices and all chemical nut graphs on up to 16 vertices. The smallest order for which a regular nut graph exists is 8; see also [8]. In [8], Fowler et al. presented the following question. Question 1. Is it true that for each d ≥ 3 there are only finitely many numbers n such that there does not exist a d-regular nut graph of order n? In the attempt to answer Question 1, the ‘Fowler Construction’ played an important role; see also [11]. This construction implies the following theorem. Theorem 2. Let G be a nut graph on n vertices and let u be a vertex of G of degree d. Then there exists a nut graph of order n + 2d that is obtained from G by adding 2d new vertices and rearranging the edges in a certain way. In the newly obtained nut graph the degrees of the new vertices are d and the degrees of the original vertices are not changed. Obviously, if G is a d-regular graph of order n, then the new graph is d-regular of order n + 2d. Hence, to positively answer the Question 1 for specific degree d, it suffices to find d-regular graphs for 2d consecutive orders. In [11] (d = 3, 4) and [8] (5 ≤ d ≤ 11) the authors found all pairs (d, n), such that d ≤ 11 and there exists a d-regular nut graph of order n. In the present paper, we extend this result to d = 12. We prove the following statement. Theorem 3. There exists a 12-regular nut graph of order n if and only if n ≥ 16. To prove the ‘positive part’ of Theorem 3, it suffices to find 12-regular nut graphs of orders n ∈ {16, 17, . . . , 39}. We present these graphs in the following section. For odd orders there is not much to say; we did a computer search and thus we provide a list of graphs that we found. However, for even orders we can say more. A graph G is called vertex-transitive if all vertices are equivalent under the action of the automorphism group Aut(G). In other words, for each pair of vertices u, v ∈ V (G) there exist an automorphism α ∈ Aut(G) such that α(u) = v. In [8], the following necessary condition for a vertex-transitive nut graph was given. Theorem 4. Let G be a vertex-transitive nut graph of degree d on n vertices. Then n and d satisfy the following conditions. Either 1. d ≡ 0 (mod 4), n ≡ 0 (mod 2) and n ≥ d + 4, or 2. d ≡ 0 (mod 2), n ≡ 0 (mod 4) and n ≥ d + 6. The existence of vertex-transitive nut graphs is interesting on its own, see [8, Question 4]. For our research it is important that, by Theorem 4, there may exist vertex-transitive 12-regular graphs of even orders n ≥ 16. We found such graphs among circulant graphs. 2 2 Results We start with the ‘negative part’ of Theorem 3. There is only one 12-regular graph of order 13, namely the complete graph K13 , and it is not a nut graph. The unique 12-regular graph of order 14 is obtained by removing a matching from K14 , and again, this graph is not a nut graph. Finally, there are 17 graphs of order 15 which are 12-regular. They are obtained by removing a 2-factor from K15 . Using the SageMath software [13] we analysed all such graphs and concluded that none of them is a nut graph. Now we turn our attention to the ‘positive part’ of Theorem 3. We start with more general results for even orders. The following lemma is in fact hidden in the text preceding Proposition 1 in [11]. We decided to present it here in a slightly more general frame together with its short proof. Lemma 5. Let G be a d-regular graph on n vertices such that its adjacency matrix A is singular. Then for every eigenvector c = (c0 , c1 , . . . , cn−1 )T corresponding to eigenvalue 0 we have n−1 X ci = 0. i=0 Proof. LetPA = (a0 , a1P , . . . , an−1 )T . Then Ac = (a0 c, a1 c, .P . . , an−1 c)T = 0, and thus n−1 n−1 However, i=0 ai c = i=0 dci , and since d > 0, we have n−1 i=0 ci = 0. Pn−1 i=0 ai c = 0. Let V = {0, 1, . . . , n−1} and let 1 ≤ a1 < a2 < · · · < at ≤ n2 . By C(n, {a1 , a2 , . . . , at }) we denote a graph on the vertex set V in which two vertices i, j ∈ V are adjacent if and only if |i − j| = ak , where 1 ≤ k ≤ t. The graph C(n, {a1 , a2 , . . . , at }) is called a circulant graph and it is regular. Its degree is 2t − 1 if at = n2 and 2t otherwise. In fact, circulant graphs are vertex-transitive since ϕ : i → i + 1 is an automorphism of C(n, {a1 , a2 , . . . , at }) (the addition is modulo n). Circulant graphs are easy to describe and easy to handle. Therefore, it would be nice if there were many nut graphs among them. We prove one positive and one negative result about circulant graphs. We start with the following lemma. Lemma 6. Let G = C(n, {a1 , a2 , . . . , at }) be a circulant nut graph, and let A be its adjacency matrix. Then (1, −1, 1, −1, . . . ) is an eigenvector corresponding to eigenvalue 0. Proof. We use the well-known fact that if b and c are eigenvectors corresponding to eigenvalue λ, then b + c is also an eigenvector corresponding to eigenvalue λ. Let b = (b0 , b1 , . . . , bn−1 )T be an eigenvector corresponding to 0. Denote b0 = p and b1 = q. Since ϕ : i → 2 − i is an automorphism of G (the addition being modulo n), there is an eigenvector c = (c0 , c1 , . . . , cn−1 )T such that c2−i = −bi , 0 ≤ i ≤ n−1. Then c1 = −b1 = −q and c2 = −b0 = −p. Since b1 + c1 = 0 and b + c is an eigenvector, we must have b + c = 0 because G is a nut graph. Hence, b2 + c2 = 0 and therefore b2 = p. Now repeating the process we get b = (p, q, p, q, . . . ). Observe that n is even by Theorem 4. Thus, by Lemma 5, we have q = −p and so (1, −1, 1, −1, . . . ) is an eigenvector corresponding to eigenvalue 0. Our negative result covers all circulant graphs of degree d ≡ 2 (mod 4). 3 Theorem 7. There is no circulant nut graph of degree d if d ≡ 2 (mod 4). Proof. Let d ≡ 2 (mod 4). Denote t = d2 . Observe that t is an odd number. By way of contradiction, assume that G = C(n, {a1 , a2 , . . . , at }) is a circulant nut graph. Then n is even by Theorem 4. Let A = (a0 , a1 , . . . , an−1 )T be the adjacency matrix of G. By Lemma 6, c = (1, −1, 1, −1, . . . )T is an eigenvector corresponding to eigenvalue 0, so that Ac = 0, and in particular a0 c = 0. However, a0 c = ca1 + ca2 + · · · + cat + cn−a1 + cn−a2 + · · · + cn−at . Since cai = cn−ai for every i, 1 ≤ i ≤ t (observe that the difference between indices ai and n − ai is even), we have a0 c = 2(ca1 + ca2 + · · · + cat ), which implies that ca1 + ca2 + · · · + cat = 0. However, sum of odd number of odd numbers cannot be an even number, a contradiction. Now we prove the positive result. Theorem 8. Let d ≡ 0 (mod 4) and let n be even. Then C(n, {1, 2, . . . , d2 }) is a nut graph if and only if d2 + 1 is coprime to n and d4 is coprime to n2 . Proof. Let t = d2 . Then t is even and the graph is G = C(n, {1, 2, . . . , t}). Let A be the adjacency matrix of G. By Lemma 6, b = (1, −1, 1, −1, . . . )T is an eigenvector of A corresponding to eigenvalue 0. Thus Ab = 0. Our aim is to show that if t + 1 is coprime to n and t n 2 is coprime to 2 , then Ac = 0 if and only if c is a multiple of b. So let Ac = 0, where c = (c0 , c1 , . . . , cn−1 )T . Let A = (a0 , a1 , . . . , an−1 )T . Then at c = c0 + c1 + · · · + ct−1 + ct+1 + ct+2 + · · · + c2t = 0, at+1 c = c1 + c2 + · · · + ct + ct+2 + ct+3 + · · · + c2t+1 = 0. Subtracting the two equations we get at c − at+1 c = c0 − ct + ct+1 − c2t+1 = 0, and analogously a2t+1 c − a2t+2 c = ct+1 − c2t+1 + c2t+2 − c3t+2 = 0. This gives c0 − ct = c2t+2 − c3t+2 , and analogously c2t+2 − c3t+2 = c4t+4 − c5t+4 , c4t+4 − c5t+4 = c6t+6 − c7t+5 , etc. So if the odd number t + 1 is coprime to even number n, we get c0 − ct = c2(t+1) − ct+2(t+1) = · · · = c2 − ct+2 , which gives c2 − c0 = ct+2 − ct , 4 and analogously we get ct+2 − ct = c2t+2 − c2t , c2t+2 − c2t = c3t+2 − c3t , Here, t and n are both even. But if t 2 is coprime to n 2 etc. then c2 − c0 = ct+2 − ct = · · · = c4 − c2 . Hence, c2 − c0 = c4 − c2 = c6 − c4 = · · · Now, if c2 > c0 then c0 < c2 < c4 < · · · < c0 , a contradiction. Analogously, if c2 < c0 then c0 > c2 > c4 > · · · > c0 , a contradiction. So c0 = c2 = · · · = cn−2 and analogously c1 = c3 = · · · = cn−1 . Hence if c0 = p, then c = (p, −p, p, −p, . . . ) by Lemma 5, and the eigenspace corresponding to eigenvalue 0 is 1-dimensional. Now suppose that t + 1 is not coprime to n. Set b = 0. We will change some entries of b. Since t + 1 is odd, there is an even k such that (t + 1)k ≡ 0 (mod n) and 1 ≤ k < n. Set b0 = 1, bt+1 = −1, b2(t+1) = 1, b3(t+1) = −1, ..., where the indices are modulo n. We have changed k entries of b and since k is even, the last changed entry has value −1. Thus some entries of b remained 0’s and nevertheless Ab = 0, since if j-th entry of ai is 1, then either (j + (t + 1))-th or (j − (t + 1))-th (modulo n) entry of ai is also 1 (while the other is 0). Hence, G is not a nut graph in this case. Finally, suppose that 2t is not coprime to n2 . Then there exist a number k such that k | 2t , k | n2 and k > 1. Again, set b = 0. We will change some entries of b. Set b0 = b2 = b4 = · · · = b2(k−2) = 1 and b2(k−1) = −(k − 1), n and repeat this pattern for all even indices of b. Since k | n2 , this pattern is repeated exactly 2k times. And since every ai contains two disjoint sets of t consecutive 1’s, we have Ab = 0. But half of the entries of b are 0’s and therefore G is not a nut graph. Observe that the only requirement for n in Theorem 8 is that n is even and n > d. However, if n = d + 2 then d2 + 1 is not coprime to n, and so n ≥ d + 4. Hence, by Theorem 8, for d = 12 the following circulant graphs are nut graphs: C(16, {1, 2, 3, 4, 5, 6}), C(20, {1, 2, 3, 4, 5, 6}), C(22, {1, 2, 3, 4, 5, 6}), C(26, {1, 2, 3, 4, 5, 6}), C(32, {1, 2, 3, 4, 5, 6}), C(34, {1, 2, 3, 4, 5, 6}), and C(38, {1, 2, 3, 4, 5, 6}). Using computer [13] we found that nut graphs are also the following graphs: C(18, {1, 2, 3, 4, 5, 8}), C(24, {1, 2, 3, 4, 5, 8}), C(30, {1, 2, 3, 4, 5, 8}), and C(36, {1, 2, 3, 4, 5, 8}). We pose the following conjecture. 5 C(28, {1, 2, 3, 4, 5, 10}), Conjecture 9. For every even n, n ≥ 16, there exist a circulant nut graph C(n, {a1 , a2 , . . . , a6 }) of degree 12. We also give a more general conjecture. Conjecture 10. For every d, where d ≡ 0 (mod 4), and for every even n, n ≥ d + 4, there exists a circulant nut graph C(n, {a1 , a2 , . . . , ad/2 }) of degree d. By Theorem 4, if n is odd then there is no vertex-transitive nut graph of order n and degree 12. In this case all graphs were found by a computer search. If G is a regular graph that contains edges u1 v1 and u2 v2 but does not contain edges u1 v2 , u2 v1 , then rewiring (i.e. removing edges u1 v1 , u2 v2 and adding edges u1 v2 , u2 v1 ) yields another regular graph. Our approach was to start with a “nice” 12-regular graph of odd order and perferm a sequence of rewirings. In this way all graphs in the Appendix were obtained. For instance, the graph on 21 vertices, whose eigenvector contains only values 1 and −2, was obtained from C(21, {1, 2, 3, 4, 5, 6}) by removing the edges (0, 16) and (2, 7) and adding the edges (0, 7) and (2, 16). For n = 13 this method seems to be too time-consuming. Acknowledgements. The work of the first author is supported in part by the Slovenian Research Agency (research program P1-0294 and research projects J1-9187, J1-1691, N1-0140 and J1-2481). The second author acknowledges partial support by Slovak research grants APVV-15-0220, APVV17-0428, VEGA 1/0142/17 and VEGA 1/0238/19. The research of the third author was partially supported by the Slovenian Research Agency (ARRS), research program P1-0383 and research project J1-1692. ORCID iD Nino Bašić https://orcid.org/0000-0002-6555-8668 Martin Knor https://orcid.org/0000-0003-3555-3994 Riste Škrekovski https://orcid.org/0000-0001-6851-3214 References [1] N. Bašić P. W. Fowler, T. Pisanski and I. Sciriha, On singular signed graphs with nullspace spanned by a full vector: Signed nut graphs, arXiv:2009.09018 [math.CO], 2020, https:// arxiv.org/abs/2009.09018v4. [2] G. Brinkmann, K. Coolsaet, J. Goedgebeur and H. Mélot, House of Graphs: a database of interesting graphs, Discrete Appl. Math. 161 (2013), 311–314, doi:10.1016/j.dam.2012.07.018. [3] A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Universitext, Springer, New York, 2012, doi:10.1007/978-1-4614-1939-6. [4] K. Coolsaet, P. W. Fowler and J. Goedgebeur, Generation and properties of nut graphs, MATCH Commun. Math. Comput. Chem. 80 (2018), 423–444. [5] K. Coolsaet, P. W. Fowler and J. Goedgebeur, Nut graphs, homepage of Nutgen, http:// caagt.ugent.be/nutgen/. 6 [6] F. R. K. Chung, Spectral Graph Theory, volume 92 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 1997. [7] D. M. Cvetković, M. Doob and H. Sachs, Spectra of Graphs: Theory and Applications, Johann Ambrosius Barth, Heidelberg, 3rd edition, 1995. [8] P. W. Fowler, J. B. Gauci, J. Goedgebeur, T. Pisanski and I. Sciriha, Existence of regular nut graphs of degree at most 11, Disc. Math. Graph Theory, 40 (2020), 533–557, doi:10.7151/dmgt.2283. [9] P. W. Fowler, B. T. Pickup, T. Z. Todorova, M. Borg and I. Sciriha, Omni-conducting and omni-insulating molecules, J. Chem. Phys. 140 (2014), 054115, doi:10.1063/1.4863559. [10] P. W. Fowler, T. Pisanski and N. Bašić, Charting the space of chemical nut graphs, MATCH Commun. Math. Comput. Chem. (2020), in press. [11] J. B. Gauci, T. Pisanski and I. Sciriha, Existence of regular nut graphs and the Fowler construction, Appl. Anal. Discrete Math. (2020), in press, doi:10.2298/aadm190517028g. [12] I. Gutman and I. Sciriha, Graphs with maximum singularity, Graph Theory Notes N. Y. 30 (1996), 17–20. [13] SageMath, the Sage Mathematics Software System (Version 9.2), The Sage Developers, 2020, https://www.sagemath.org. [14] I. Sciriha, On the construction of graphs of nullity one, Discrete Math. 181 (1998), 193–211, doi:10.1016/s0012-365x(97)00036-8. [15] I. Sciriha, A characterization of singular graphs, Electron. J. Linear Algebra 16 (2007), 451–462, doi:10.13001/1081-3810.1215. [16] I. Sciriha, Coalesced and embedded nut graphs in singular graphs, Ars Math. Contemp. 1 (2008), 20–31, doi:10.26493/1855-3974.20.7cc. [17] I. Sciriha, Graphs with a common eigenvalue deck, Linear Algebra Appl. 430 (2009), 78–85, doi:10.1016/j.laa.2008.06.033. [18] I. Sciriha, Maximal core size in singular graphs, Ars Math. Contemp. 2 (2009), 217–229, doi:10.26493/1855-3974.115.891. [19] I. Sciriha and P. W. Fowler, Nonbonding orbitals in fullerenes: Nuts and cores in singular polyhedral graphs, J. Chem. Inf. Model. 47 (2007), 1763–1775, doi:10.1021/ci700097j. [20] I. Sciriha and I. Gutman, Nut graphs: Maximally extending cores, Util. Math. 54 (1998), 257–272. 7 Appendix A 12-regular nut graphs of odd orders Here, we list one 12-regular nut graph of odd order n for each n ∈ {17, 19, . . . , 39}. Each graph is given in the adjacency-lists (of neighbours of each vertex) representaion, formatted as a Python dictionary. We also give the corresponding kernel eigenvector c as a list of integer entries. Order n = 17. {0: [1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 15, 16], 1: [0, 2, 3, 4, 6, 7, 8, 9, 10, 11, 15, 16], 2: [0, 1, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15], 3: [0, 1, 4, 6, 7, 8, 9, 11, 12, 14, 15, 16], 4: [0, 1, 2, 3, 5, 6, 8, 9, 10, 11, 13, 16], 5: [0, 2, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15], 6: [1, 2, 3, 4, 5, 7, 8, 9, 12, 13, 14, 15], 7: [1, 2, 3, 5, 6, 8, 10, 11, 12, 13, 14, 16], 8: [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 13, 14], 9: [0, 1, 2, 3, 4, 5, 6, 10, 12, 13, 14, 16], 10: [0, 1, 2, 4, 5, 7, 8, 9, 12, 14, 15, 16], 11: [0, 1, 2, 3, 4, 7, 8, 12, 13, 14, 15, 16], 12: [0, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16], 13: [2, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 16], 14: [3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16], 15: [0, 1, 2, 3, 5, 6, 10, 11, 12, 13, 14, 16], 16: [0, 1, 3, 4, 7, 9, 10, 11, 12, 13, 14, 15]} c = [3, −3, −2, 2, 1, 2, −1, −2, 3, −1, −1, 1, 1, −1, 1, −1, −2] Order n = 19. {0: [1, 2, 5, 7, 9, 10, 11, 12, 13, 14, 16, 18], 1: [0, 3, 5, 6, 7, 10, 12, 13, 14, 15, 17, 18], 2: [0, 4, 6, 7, 8, 9, 10, 11, 12, 16, 17, 18], 3: [1, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18], 4: [2, 5, 6, 7, 8, 11, 12, 13, 14, 15, 17, 18], 5: [0, 1, 4, 7, 8, 9, 11, 12, 13, 14, 15, 17], 6: [1, 2, 3, 4, 7, 8, 9, 10, 14, 15, 16, 17], 7: [0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 15, 16], 8: [2, 3, 4, 5, 6, 7, 9, 11, 14, 15, 17, 18], 9: [0, 2, 5, 6, 7, 8, 10, 11, 12, 13, 16, 17], 10: [0, 1, 2, 3, 6, 9, 11, 12, 13, 14, 16, 18], 11: [0, 2, 3, 4, 5, 7, 8, 9, 10, 16, 17, 18], 12: [0, 1, 2, 3, 4, 5, 9, 10, 13, 14, 15, 16], 13: [0, 1, 3, 4, 5, 9, 10, 12, 14, 15, 16, 17], 14: [0, 1, 3, 4, 5, 6, 8, 10, 12, 13, 15, 18], 15: [1, 4, 5, 6, 7, 8, 12, 13, 14, 16, 17, 18], 16: [0, 2, 3, 6, 7, 9, 10, 11, 12, 13, 15, 18], 17: [1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 18], 18: [0, 1, 2, 3, 4, 8, 10, 11, 14, 15, 16, 17]} c = [5, 10, 6, −10, −3, −1, 4, −1, −5, 1, 1, −5, −4, −3, −4, 2, −4, 7, 4] Order n = 21. {0: [1, 2, 3, 4, 5, 6, 7, 15, 17, 18, 19, 20], 1: [0, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20], 2: [0, 1, 3, 4, 5, 6, 8, 16, 17, 18, 19, 20], 3: [0, 1, 2, 4, 5, 6, 7, 8, 9, 18, 19, 20], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 19, 20], 5: [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 20], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [0, 1, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13], 8: [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15], 10: [4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16], 11: [5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17], 12: [6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19], 14: [8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20], 15: [0, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20], 16: [1, 2, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20], 17: [0, 1, 2, 11, 12, 13, 14, 15, 16, 18, 19, 20], 18: [0, 1, 2, 3, 12, 13, 14, 15, 16, 17, 19, 20], 19: [0, 1, 2, 3, 4, 13, 14, 15, 16, 17, 18, 20], 20: [0, 1, 2, 3, 4, 5, 14, 15, 16, 17, 18, 19]} c = [1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1] Order n = 23. {0: [1, 2, 4, 6, 7, 8, 10, 11, 13, 19, 20, 21], 1: [0, 4, 5, 6, 7, 9, 11, 13, 16, 17, 20, 22], 2: [0, 3, 4, 6, 8, 8 11, 12, 13, 16, 19, 20, 21], 3: [2, 4, 5, 8, 9, 10, 12, 13, 14, 16, 17, 18], 4: [0, 1, 2, 3, 6, 7, 8, 14, 15, 16, 21, 22], 5: [1, 3, 7, 10, 11, 12, 14, 15, 17, 18, 19, 20], 6: [0, 1, 2, 4, 11, 12, 14, 17, 18, 19, 20, 22], 7: [0, 1, 4, 5, 10, 11, 12, 16, 18, 19, 21, 22], 8: [0, 2, 3, 4, 9, 10, 12, 13, 15, 16, 21, 22], 9: [1, 3, 8, 10, 11, 13, 14, 15, 18, 19, 21, 22], 10: [0, 3, 5, 7, 8, 9, 11, 13, 17, 18, 19, 21], 11: [0, 1, 2, 5, 6, 7, 9, 10, 13, 14, 15, 20], 12: [2, 3, 5, 6, 7, 8, 13, 14, 17, 19, 20, 22], 13: [0, 1, 2, 3, 8, 9, 10, 11, 12, 14, 15, 19], 14: [3, 4, 5, 6, 9, 11, 12, 13, 15, 16, 17, 20], 15: [4, 5, 8, 9, 11, 13, 14, 17, 18, 20, 21, 22], 16: [1, 2, 3, 4, 7, 8, 14, 17, 18, 20, 21, 22], 17: [1, 3, 5, 6, 10, 12, 14, 15, 16, 19, 20, 21], 18: [3, 5, 6, 7, 9, 10, 15, 16, 19, 20, 21, 22], 19: [0, 2, 5, 6, 7, 9, 10, 12, 13, 17, 18, 22], 20: [0, 1, 2, 5, 6, 11, 12, 14, 15, 16, 17, 18], 21: [0, 2, 4, 7, 8, 9, 10, 15, 16, 17, 18, 22], 22: [1, 4, 6, 7, 8, 9, 12, 15, 16, 18, 19, 21]} c = [6, −24, −7, 13, 39, 1, 27, 4, −18, −4, 10, 3, −14, −14, 28, 1, −22, −2, 3, 6, −28, 2, −10] Order n = 25. {0: [3, 4, 5, 7, 9, 10, 12, 13, 17, 19, 22, 23], 1: [2, 3, 5, 11, 12, 15, 16, 18, 19, 20, 21, 23], 2: [1, 3, 4, 5, 10, 13, 14, 17, 20, 21, 23, 24], 3: [0, 1, 2, 5, 8, 10, 14, 16, 20, 21, 23, 24], 4: [0, 2, 6, 8, 9, 10, 11, 13, 18, 21, 23, 24], 5: [0, 1, 2, 3, 10, 13, 14, 17, 18, 19, 20, 24], 6: [4, 8, 9, 10, 11, 12, 14, 17, 19, 20, 21, 22], 7: [0, 8, 9, 11, 12, 15, 16, 18, 19, 22, 23, 24], 8: [3, 4, 6, 7, 9, 10, 11, 13, 17, 18, 22, 23], 9: [0, 4, 6, 7, 8, 10, 11, 12, 14, 15, 18, 21], 10: [0, 2, 3, 4, 5, 6, 8, 9, 15, 16, 17, 18], 11: [1, 4, 6, 7, 8, 9, 12, 13, 14, 17, 19, 20], 12: [0, 1, 6, 7, 9, 11, 13, 14, 15, 18, 21, 22], 13: [0, 2, 4, 5, 8, 11, 12, 16, 20, 21, 22, 23], 14: [2, 3, 5, 6, 9, 11, 12, 15, 16, 17, 19, 22], 15: [1, 7, 9, 10, 12, 14, 16, 17, 19, 20, 22, 24], 16: [1, 3, 7, 10, 13, 14, 15, 17, 18, 19, 20, 24], 17: [0, 2, 5, 6, 8, 10, 11, 14, 15, 16, 21, 23], 18: [1, 4, 5, 7, 8, 9, 10, 12, 16, 21, 22, 24], 19: [0, 1, 5, 6, 7, 11, 14, 15, 16, 21, 22, 24], 20: [1, 2, 3, 5, 6, 11, 13, 15, 16, 22, 23, 24], 21: [1, 2, 3, 4, 6, 9, 12, 13, 17, 18, 19, 23], 22: [0, 6, 7, 8, 12, 13, 14, 15, 18, 19, 20, 24], 23: [0, 1, 2, 3, 4, 7, 8, 13, 17, 20, 21, 24], 24: [2, 3, 4, 5, 7, 15, 16, 18, 19, 20, 22, 23]} c = [29, 20, −31, 7, 5, −13, 32, −19, −12, 1, 31, −12, −8, −6, −49, 17, 3, −17, −21, 20, 33, 7, 1, −2, −16] Order n = 27. {0: [2, 3, 4, 5, 6, 7, 21, 22, 23, 24, 25, 26], 1: [2, 3, 4, 5, 6, 7, 8, 22, 23, 24, 25, 26], 2: [0, 1, 3, 4, 5, 6, 7, 8, 23, 24, 25, 26], 3: [0, 1, 2, 4, 5, 6, 7, 8, 9, 24, 25, 26], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 25, 26], 5: [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 26], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13], 8: [1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15], 10: [4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16], 11: [5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17], 12: [6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19], 14: [8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20], 15: [9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21], 16: [10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22], 17: [11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23], 18: [12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24], 19: [13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25], 20: [14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26], 21: [0, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26], 22: [0, 1, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26], 23: [0, 1, 2, 17, 18, 19, 20, 21, 22, 24, 25, 26], 24: [0, 1, 2, 3, 18, 19, 20, 21, 22, 23, 25, 26], 25: [0, 1, 2, 3, 4, 19, 20, 21, 22, 23, 24, 26], 26: [0, 1, 2, 3, 4, 5, 20, 21, 22, 23, 24, 25]} c = [1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1] 9 Order n = 29. {0: [1, 3, 5, 6, 9, 10, 11, 13, 14, 19, 26, 28], 1: [0, 2, 4, 5, 11, 16, 17, 18, 19, 21, 26, 27], 2: [1, 3, 8, 9, 10, 11, 13, 24, 25, 26, 27, 28], 3: [0, 2, 12, 13, 17, 20, 21, 23, 24, 25, 26, 27], 4: [1, 5, 6, 9, 11, 15, 16, 17, 20, 22, 23, 28], 5: [0, 1, 4, 7, 12, 15, 16, 19, 20, 22, 24, 25], 6: [0, 4, 7, 8, 9, 11, 15, 17, 18, 19, 21, 22], 7: [5, 6, 8, 11, 12, 13, 15, 16, 18, 20, 22, 24], 8: [2, 6, 7, 10, 12, 15, 19, 20, 21, 24, 26, 27], 9: [0, 2, 4, 6, 12, 14, 15, 20, 22, 23, 24, 27], 10: [0, 2, 8, 13, 16, 17, 18, 20, 21, 23, 25, 26], 11: [0, 1, 2, 4, 6, 7, 12, 16, 17, 19, 20, 23], 12: [3, 5, 7, 8, 9, 11, 14, 15, 18, 19, 21, 25], 13: [0, 2, 3, 7, 10, 14, 15, 21, 23, 25, 27, 28], 14: [0, 9, 12, 13, 15, 18, 22, 23, 24, 26, 27, 28], 15: [4, 5, 6, 7, 8, 9, 12, 13, 14, 18, 22, 27], 16: [1, 4, 5, 7, 10, 11, 18, 20, 21, 25, 27, 28], 17: [1, 3, 4, 6, 10, 11, 18, 19, 22, 24, 27, 28], 18: [1, 6, 7, 10, 12, 14, 15, 16, 17, 19, 23, 24], 19: [0, 1, 5, 6, 8, 11, 12, 17, 18, 23, 26, 27], 20: [3, 4, 5, 7, 8, 9, 10, 11, 16, 25, 26, 28], 21: [1, 3, 6, 8, 10, 12, 13, 16, 22, 23, 25, 26], 22: [4, 5, 6, 7, 9, 14, 15, 17, 21, 24, 25, 27], 23: [3, 4, 9, 10, 11, 13, 14, 18, 19, 21, 24, 28], 24: [2, 3, 5, 7, 8, 9, 14, 17, 18, 22, 23, 28], 25: [2, 3, 5, 10, 12, 13, 16, 20, 21, 22, 26, 28], 26: [0, 1, 2, 3, 8, 10, 14, 19, 20, 21, 25, 28], 27: [1, 2, 3, 8, 9, 13, 14, 15, 16, 17, 19, 22], 28: [0, 2, 4, 13, 14, 16, 17, 20, 23, 24, 25, 26]} c = [1, 1, 37, −13, −20, −42, 21, −5, −36, 25, 5, 30, 41, −25, 21, −6, 6, 17, 34, −34, −14, −13, 7, −51, −16, 39, 5, −21, 6] Order n = 31. {0: [5, 10, 12, 13, 17, 18, 21, 22, 24, 26, 27, 29], 1: [3, 6, 7, 8, 10, 14, 17, 20, 23, 25, 27, 30], 2: [4, 7, 9, 10, 18, 21, 22, 23, 24, 25, 27, 28], 3: [1, 4, 5, 11, 13, 16, 17, 18, 19, 24, 25, 29], 4: [2, 3, 5, 11, 12, 13, 18, 21, 25, 26, 28, 29], 5: [0, 3, 4, 6, 7, 9, 11, 14, 17, 25, 27, 29], 6: [1, 5, 8, 9, 11, 13, 18, 20, 22, 26, 29, 30], 7: [1, 2, 5, 9, 10, 12, 20, 24, 25, 26, 27, 30], 8: [1, 6, 9, 14, 15, 17, 18, 20, 21, 22, 23, 30], 9: [2, 5, 6, 7, 8, 12, 14, 15, 19, 24, 27, 28], 10: [0, 1, 2, 7, 12, 13, 15, 18, 19, 21, 24, 28], 11: [3, 4, 5, 6, 12, 15, 17, 20, 22, 23, 29, 30], 12: [0, 4, 7, 9, 10, 11, 14, 16, 18, 21, 27, 30], 13: [0, 3, 4, 6, 10, 16, 20, 23, 24, 25, 26, 27], 14: [1, 5, 8, 9, 12, 15, 17, 18, 19, 20, 22, 23], 15: [8, 9, 10, 11, 14, 17, 19, 20, 21, 27, 28, 30], 16: [3, 12, 13, 18, 19, 21, 22, 23, 24, 26, 28, 29], 17: [0, 1, 3, 5, 8, 11, 14, 15, 20, 22, 23, 29], 18: [0, 2, 3, 4, 6, 8, 10, 12, 14, 16, 24, 25], 19: [3, 9, 10, 14, 15, 16, 20, 21, 22, 23, 26, 28], 20: [1, 6, 7, 8, 11, 13, 14, 15, 17, 19, 24, 25], 21: [0, 2, 4, 8, 10, 12, 15, 16, 19, 25, 27, 29], 22: [0, 2, 6, 8, 11, 14, 16, 17, 19, 23, 28, 30], 23: [1, 2, 8, 11, 13, 14, 16, 17, 19, 22, 26, 28], 24: [0, 2, 3, 7, 9, 10, 13, 16, 18, 20, 28, 30], 25: [1, 2, 3, 4, 5, 7, 13, 18, 20, 21, 26, 29], 26: [0, 4, 6, 7, 13, 16, 19, 23, 25, 27, 29, 30], 27: [0, 1, 2, 5, 7, 9, 12, 13, 15, 21, 26, 30], 28: [2, 4, 9, 10, 15, 16, 19, 22, 23, 24, 29, 30], 29: [0, 3, 4, 5, 6, 11, 16, 17, 21, 25, 26, 28], 30: [1, 6, 7, 8, 11, 12, 15, 22, 24, 26, 27, 28]} c = [1, 91, −39, 14, 39, 33, 75, −48, −37, 2, 146, −14, −13, 23, 20, 6, −84, −32, 27, 38, −93, −66, −43, 21, −79, −43, 18, −15, 59, 1, −8] Order n = 33. {0: [1, 2, 3, 4, 5, 6, 27, 28, 29, 30, 31, 32], 1: [0, 2, 3, 4, 5, 6, 7, 11, 28, 29, 31, 32], 2: [0, 1, 3, 4, 5, 6, 7, 8, 29, 30, 31, 32], 3: [0, 1, 2, 4, 5, 6, 7, 8, 9, 30, 31, 32], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 31, 32], 5: [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 32], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13], 8: [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15], 10: [4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 30], 11: [1, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17], 12: [6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19], 14: [8, 9, 10, 11, 12, 13, 15, 10 16, 18, 20, 22, 24, 26, 28, 29, 31, 17, 18, 19, 20], 15: [9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21], 16: [10, 11, 12, 13, 14, 15, 19, 20, 21, 22], 17: [11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23], 18: [12, 13, 14, 15, 16, 17, 21, 22, 23, 24], 19: [13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25], 20: [14, 15, 16, 17, 18, 19, 23, 24, 25, 26], 21: [15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27], 22: [16, 17, 18, 19, 20, 21, 25, 26, 27, 28], 23: [17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29], 24: [18, 19, 20, 21, 22, 23, 27, 28, 29, 30], 25: [19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31], 26: [20, 21, 22, 23, 24, 25, 29, 30, 31, 32], 27: [0, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32], 28: [0, 1, 22, 23, 24, 25, 26, 30, 31, 32], 29: [0, 1, 2, 23, 24, 25, 26, 27, 28, 30, 31, 32], 30: [0, 2, 3, 10, 24, 25, 26, 27, 28, 32], 31: [0, 1, 2, 3, 4, 25, 26, 27, 28, 29, 30, 32], 32: [0, 1, 2, 3, 4, 5, 26, 27, 28, 29, 30, 31]} 17, 19, 21, 23, 25, 27, 27, 29, c = [1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1] Order n = 35. {0: [1, 2, 3, 4, 5, 6, 29, 30, 31, 32, 33, 34], 1: [0, 2, 3, 4, 5, 6, 7, 30, 31, 32, 33, 34], 2: [0, 1, 3, 4, 5, 6, 7, 8, 15, 31, 32, 33], 3: [0, 1, 2, 4, 5, 6, 8, 9, 15, 32, 33, 34], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 31, 33, 34], 5: [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 34], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [1, 2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 21], 8: [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15], 10: [5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 25], 11: [5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17], 12: [6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 34], 14: [8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20], 15: [2, 3, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20], 16: [10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22], 17: [11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23], 18: [12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24], 19: [13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25], 20: [14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26], 21: [7, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27], 22: [16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28], 23: [17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29], 24: [18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30], 25: [10, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30], 26: [20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32], 27: [21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33], 28: [22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34], 29: [0, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34], 30: [0, 1, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34], 31: [0, 1, 2, 4, 26, 27, 28, 29, 30, 32, 33, 34], 32: [0, 1, 2, 3, 26, 27, 28, 29, 30, 31, 33, 34], 33: [0, 1, 2, 3, 4, 27, 28, 29, 30, 31, 32, 34], 34: [0, 1, 3, 4, 5, 13, 28, 29, 30, 31, 32, 33]} c = [1, −1, −1, −3, 3, 2, −1, −1, 1, 1, −2, 2, −2, −1, 3, −1, −1, 2, −2, −2, 5, −1, −1, 1, −2, −2, 6, −3, −1, 1, −1, 5, −1, −4, 1] Order n = 37. {0: [1, 2, 3, 4, 5, 6, 31, 32, 33, 34, 35, 36], 1: [0, 2, 3, 4, 5, 6, 7, 18, 22, 32, 33, 35], 2: [0, 1, 3, 4, 5, 6, 7, 8, 33, 34, 35, 36], 3: [0, 1, 2, 4, 5, 6, 7, 8, 9, 34, 35, 36], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 35, 36], 5: [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 36], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13], 8: [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 15, 32], 10: [4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16], 11: [5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17], 12: [6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 19, 36], 14: [8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 35], 15: [9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21], 16: [10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22], 17: [11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23], 18: [1, 12, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24], 19: [13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25], 20: [14, 15, 16, 17, 18, 19, 11 21, 21, 25, 27, 29, 31, 30, 33, 22, 23, 24, 25, 26], 21: [15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27], 22: [1, 16, 17, 18, 19, 20, 23, 24, 26, 27, 28], 23: [17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29], 24: [18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30], 25: [19, 20, 21, 23, 24, 26, 27, 28, 29, 30, 31, 34], 26: [20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32], 27: [21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33], 28: [22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34], 29: [23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35], 30: [24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36], 31: [0, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36], 32: [0, 1, 9, 26, 27, 28, 29, 31, 33, 34, 36], 33: [0, 1, 2, 27, 28, 29, 30, 31, 32, 34, 35, 36], 34: [0, 2, 3, 25, 28, 29, 30, 31, 32, 35, 36], 35: [0, 1, 2, 3, 4, 14, 29, 30, 31, 33, 34, 36], 36: [0, 2, 3, 4, 5, 13, 30, 31, 32, 33, 34, 35]} c = [2, −3, −4, 5, 1, −1, −1, −4, 5, 2, −5, 1, 1, −1, 6, −5, −4, 7, −1, −5, 4, −5, 3, 6, −5, −5, 8, −3, 1, 1, −4, 3, 4, −7, −1, 3, 1] Order n = 39. {0: [1, 2, 3, 4, 5, 6, 15, 33, 34, 36, 37, 38], 1: [0, 2, 3, 4, 5, 6, 7, 34, 35, 36, 37, 38], 2: [0, 1, 3, 4, 5, 6, 7, 8, 35, 36, 37, 38], 3: [0, 1, 2, 4, 5, 6, 7, 8, 9, 36, 37, 38], 4: [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 37, 38], 5: [0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 38], 6: [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12], 7: [1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13], 8: [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14], 9: [3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15], 10: [4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16], 11: [5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 17, 35], 12: [6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18], 13: [7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19], 14: [8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20], 15: [0, 9, 10, 12, 13, 14, 16, 17, 18, 19, 20, 21], 16: [10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22], 17: [11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23], 18: [12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24], 19: [13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25], 20: [14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26], 21: [15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27], 22: [16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28], 23: [17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29], 24: [18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30], 25: [19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31], 26: [20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32], 27: [21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33], 28: [22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34], 29: [23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35], 30: [24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36], 31: [25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37], 32: [26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38], 33: [0, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38], 34: [0, 1, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38], 35: [1, 2, 11, 29, 30, 31, 32, 33, 34, 36, 37, 38], 36: [0, 1, 2, 3, 30, 31, 32, 33, 34, 35, 37, 38], 37: [0, 1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 38], 38: [0, 1, 2, 3, 4, 5, 32, 33, 34, 35, 36, 37]} c = [1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1, 1, −2, 1] 12