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Properties of Entropy-Based Topological Measures of Fullerenes

Modjtaba Ghorbani

^1,*

Matthias Dehmer

^2,* and

Frank Emmert-Streib

^3,4

Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Lavizan, Tehran 16785-163, Iran

Department of Computer Science, Swiss Distance University of Applied Sciences, 3900 Brig, Switzerland

Predictive Society and Data Analytics Lab, Tampere University, Tampere, Korkeakoulunkatu 10, 33720 Tampere, Finland

⁴

Institute of Biosciences and Medical Technology, Tampere University, Tampere, Korkeakoulunkatu 10, 33720 Tampere, Finland

Authors to whom correspondence should be addressed.

Mathematics 2020, 8(5), 740; https://doi.org/10.3390/math8050740

Submission received: 8 April 2020 / Revised: 19 April 2020 / Accepted: 20 April 2020 / Published: 7 May 2020

(This article belongs to the Special Issue Graph Theory at Work in Carbon Chemistry)

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Browse Figures

Figure 1
Illustration of two cubic graphs: (a) Cuneane; (b) cube. "> Figure 2
The 2-D graph of zig-zag nanotube TZ[5,10]. "> Figure 3
Cap B. "> Figure 4
The presentation of fullerene A10n as a combining of two copies of B and the nanotube TZ[5,n − 4]. "> Figure 5
Three-dimensional graph of zig-zag polyhex nanotori. "> Figure 6
Two-dimensional lattice for T[p,q]. "> Figure 7
The molecular graph of the fullerene C24n+12, for n = 3. "> Figure 8
The molecular graph of the fullerene C12n+2. "> Figure 9
Two vertices with the same total distance which are not H-equivalent. "> Figure 10
The vertices u and v are not H-equivalent but D(u) = D(v). "> Figure 11
A cubic co-distance graph of diameter 4 with non-zero H-entropy. "> Figure 12
The fullerene A12n+4. "> Figure 13
The orbits of the ith layer (2 ≤ i ≤ n)of the fullerene graph A12n+4. "> Figure 14
The subgraph B1. "> Figure 15
The subgraph B2. "> Figure 16
The 3-dimensional structure of fullerene graph A12n+4. "> Figure 17
The Hosoya-partitions of Tz[6,n − 10]. ">

Versions Notes

Abstract

A fullerene is a cubic three-connected graph whose faces are entirely composed of pentagons and hexagons. Entropy applied to graphs is one of the significant approaches to measuring the complexity of relational structures. Recently, the research on complex networks has received great attention, because many complex systems can be modelled as networks consisting of components as well as relations among these components. Information—theoretic measures have been used to analyze chemical structures possessing bond types and hetero-atoms. In the present article, we reviewed various entropy-based measures on fullerene graphs. In particular, we surveyed results on the topological information content of a graph, namely the orbit-entropy I_a(G), the symmetry index, a degree-based entropy measure I_λ(G), the eccentric-entropy If_σ(G) and the Hosoya entropy H(G).

Keywords:

fullerene; graph entropy; automorphism group; eigenvalue; eccentricity

1. Introduction

In recent years, research on complex networks has been essential since many complex systems are modelled as networks consisting of components as well as relations among these components. Some studies focused on finding properties of real networks, such as degree distribution [1,2], degree correlation [3,4,5], and degree-based structure entropies [6,7]. Many significant properties of systems such as heterogeneity [1], assortative mixing [8,9], and self-similarity [10,11,12] are based on these statistics.

Entropy applied to graphs is one of two major approaches to measuring the complexity of relational structures. The origin of such measures goes back to Rashevsky who introduced the concept of topological information content; see [13]. Studies of graph complexity have been performed in many areas [8,11,14,15], such as chemistry to labeled chemical structures possessing bond types and hetero-atoms; see [16]. The aim of this paper was to review some selected entropy-based measures.

2. Definitions and Preliminaries

Here, we recall the definition of the automorphism group of a graph. It is a well-known fact that in a regular polyhedral graph such as a fullerene, the symmetry group and the automorphism group are the same; see [17]. A permutation π on the set of vertices of graph G, which preserves the adjacency of vertices of G, is called an automorphism. In other words, the permutation π is an automorphism if

u v \in E (G) \Leftrightarrow π (u) π (v) \in E (G) .

The set of all automorphisms of G, denoted by

Aut (G)

, forms a group under the composition of mappings called an automorphism group. Let the automorphism group Aut (G) act on vertex set V. This action yields an automorphism partition P = {V₁, V₂,..., V_k} in the which two vertices x and y are equivalent if and only if there exists g in Aut (G) such that xg = y. Each member of P is an orbit of Aut (G). The set of orbits of a graph enables us to investigate the heterogeneity of networks.

We say that Aut (G) acts transitively on the set of vertices, if for

u, v \in V (G)

, there is an automorphism

β \in Aut (G)

, such that

β (u) = v .

In this case, we say that G is vertex-transitive. It is not difficult to see that every vertex-transitive graph has only one orbit whose size is equal to the number of vertices. For example, consider the rotation ρ =

{(360 / n)}^{\circ}

of cycle graph C_n; then, the corresponding permutation is

ρ = (1, 2, \dots, n)

which yields that C_n is vertex-transitive.

Measuring the heterogeneity of complex networks has been important in studies of the behaviour of complex networks. For example, existing heterogeneity measures [6,7] of complex networks are based on degree. Specifically, entropies in [6,7] are based on degree distribution [8,9]. As is shown in Figure 1, in a network, vertices with the same degree can be distinguished by measurement on some structural properties of particular vertex such as the number of triangles that a vertex lie on or the shortest path passing through a vertex; see Figure 1.

The molecular graph of cuneane is depicted in Figure 1a. This graph is not vertex-transitive. The 2-D graph of the cube graph Q₃, as depicted in Figure 1b, indicates an example of a vertex-transitive graph. One can easily verify that the vertex partition P = {{1, 8}, {4, 5}, {2, 3, 6, 7}} in the cuneane graph is finer than the degree partition. Furthermore, we can validate that partition P is an orbit decomposition of cuneane.

For most of networks, the orbit-partition of a graph is much finer than the degree partition. The number of automorphism partitions or the number of orbits can interpret the structure of a network. For example, a graph with n vertices has exactly n orbits if and only if it is asymmetric.

3. Fullerene Graphs

In addition to the two popular forms of carbon, namely diamond and graphite, a third form of carbon called fullerene was discovered in 1985; see [18,19]. One of the most well-known members of this class of molecular graphs, the buckminster fullerene C₆₀ which contains 60 carbon atoms, 12 pentagons and 20 hexagons, is vertex-transitive.

In general, a fullerene is a cubic three-connected graph on n vertices which has n/2 - 10 hexagons and 12 pentagons.

The non-classical fullerenes are formed by a combination of triangles, quadrangles, pentagons, hexagons, etc. There are many problems concerning fullerene graphs, and many properties of them were studied by mathematicians, see [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64] as well as [65,66,67,68,69,70,71,72,73,74,75]. Fullerenes are individual cases of a larger class of graphs, namely polyhedral graphs. In general, a polyhedral graph is a three-connected simple planar graph and the polyhedral graphs considered in this paper are cubic. In chemistry, a fullerene is a molecule composed of carbon atoms in the form of many shapes such as a hollow sphere, ellipsoid, tube, etc. [51]. The non-classical fullerenes may contain hexagons and other rings.

In [76], a method is described to obtain a fullerene graph from a zig-zag or armchair nanotubes. Here, by continuing that method, we construct some infinite classes of fullerenes. Denoted by T_Z[m,n] means a zig-zag nanotube with m rows and n columns of hexagons, see Figure 2. Combine a nanotube T_Z[5,n − 4] with two copies of caps B (Figure 3), to construct a fullerene graph as shown in Figure 4. Two caps have together 40 vertices and thus, the number of vertices of the fullerene graph is 10(n − 4) + 40 = 10n. This is why we did it by A_10n.

4. Entropy Measure

The concepts of entropies have been investigated extensively in [77,78,79,80,81,82,83,84] to characterize and quantify the structure of networks. Several types of them are discussed in [1], and mathematically explored. Therefore, we omit an extensive review on graph entropies here. Studying the entropy of dendrimers, a graph class with a long-standing history in chemistry and related disciplines, is done in some references such as [82,83,84] in which authors considered the individual eccentricity-based information functionals. As graph entropies, we use a specific definition due to Dehmer [85,86,87,88,89,90,91,92] as follows. Consider a probability vector p = (p₁,…,p_n) that satisfies in two conditions 0 ≤ p_i ≤ 1 and

\sum_{i = 1}^{n} p_{i} = 1

. The Shannon’s entropy is

I (p) = - \sum_{i = 1}^{n} p_{i} \log p_{i}

, where the symbol “log” is the logarithm on the basis 2. Let

p_{i} = λ_{i} / \sum_{j = 1}^{n} λ_{j}, (i = 1, 2, \dots, n),

where

(λ_{1}, λ_{2}, \dots, λ_{n})

is a tuple of none-negative integers,

λ_{i} \in ℕ

, see [90]. The entropy of tuple

(λ_{1}, λ_{2}, \dots, λ_{n})

is given by

I (λ_{1}, λ_{2}, \dots, λ_{n}) = \log (\sum_{i = 1}^{n} λ_{i}) - \sum_{i = 1}^{n} (λ_{i} / \sum_{j = 1}^{n} λ_{j}) \log λ_{i} .

There are many ways to obtain the tuple

(λ_{1}, λ_{2}, \dots, λ_{n})

For a vertex v_i of graph G, if we put

p (v_{i}) : = f (v_{i}) / \sum_{j = 1}^{| V |} f (v_{j}),

(1)

Then clearly,

\sum_{j = 1}^{| V |} p (v_{j}) = 1

and the entropy measure of G based on f is thus

I_{f} (G) = - \sum_{i = 1}^{| V |} \frac{f (v_{i})}{\sum_{j = 1}^{| V |} f (v_{j})} \log (f (v_{i}) / \sum_{j = 1}^{| V |} f (v_{j})) = \log (\sum_{i = 1}^{| V |} f (v_{i})) - \sum_{i = 1}^{| V |} (f (v_{i}) / \sum_{j = 1}^{| V |} f (v_{j})) \log f (v_{i}) .

In the literature, there are various ways to obtain the tuple (p₁,…,p_n) like the so-called magnitude-based information measures introduced by Bonchev and Trinajstić [82], or partition—independent graph entropies, introduced by Dehmer based on information functionals; see [91,92,93,94,95,96] as well as [97,98,99,100,101,102,103,104,105,106].

A function

d : G \times G \to ℝ^{+}

in which for any pair of vertices of graph G such as x and y, d(x,y) is defined as the length of shortest path connecting them, indicates the distance between these vertices.

A topological index of graph G is a numerical quantity, which is invariant under its automorphism group. The Wiener index is the first reported distance based topological index, and it defined as half sum of the distances between all the pairs of vertices in a molecular graph; see [107]. In other words,

W (G) = \frac{1}{2} \sum_{x, y \in V (G)} d (x, y) .

(2)

The Wiener (or Hosoya) polynomial of a graph [108] is defined as

H (G, x) = \sum_{u v \in E (G)} x^{d (u, v)} .

(3)

It is clear that H(G,1) = W(G). For a given vertex v, the function

H_{v} (G, x) = \sum_{u \in V (G), u \neq v} x^{d (u, v)}

is called the partial Hosoya polynomial of G at vertex v.

Hence,

H (G, x) = \sum_{v \in V (G)} H_{v} (G, x)

(4)

Graph entropy measures represent information—theoretic measures for characterizing networks quantitatively. The first concepts in this framework was developed in the 1950s for investigating biological and chemical systems. Seminal work on this problem was done by Rashevsky, Trucco, and Mowshowitz, [100,101,102,103,104,109,110,111,112] who studied entropy measures for determining the structural information content of a graph. Graph entropies have been applied to various problems such as biology, computational biology, mathematical chemistry, Web mining, and knowledge engineering.

4.1. Eccentric Entropy Measure

The eccentricity of vertex v is an information functional defined by

σ (v) = \max_{u \in V} d (u, v)

, see [113,114,115]. Now, for a vertex

v_{i} \in V

f(v_i):= c_iσ(v_i), where c_i > 0 for 1 ≤ i ≤ n. Then

I f_{σ} (G) = \log (\sum_{i = 1}^{n} c_{i} σ (v_{i})) - \sum_{i = 1}^{n} \frac{c_{i} σ (v_{i})}{\sum_{j = 1}^{n} c_{j} σ (v_{j})} \log (c_{i} σ (v_{i})) .

Moreover, if c_is are equal, then

I f_{σ} (G) = \log (\sum_{i = 1}^{n} σ (v_{i})) - \sum_{i = 1}^{n} \frac{σ (v_{i})}{\sum_{j = 1}^{n} σ (v_{j})} \log (σ (v_{i})) .

Theorem 1

([25]). In a vertex-transitive graph, all vertices have the same eccentricity.

Theorem 2.

If G is a regular graph on n vertices, then

I f_{σ} (G) = \log (\sum_{i = 1}^{n} c_{i}) - \sum_{i = 1}^{n} \frac{c_{i}}{\sum_{j = 1}^{n} c_{j}} \log (c_{i}) .

(5)

In particular, if c_i = c_j for all, i ≠ j then If_σ(G) = log(n).

Proof.

By Theorem 1, for all

x, y \in V (G)

, we have σ(x) = σ(y) which proves the first claim.

If c_i = c_j for all i ≠ j then we have

I f_{σ} (G) = \log (n c_{1}) - \frac{1}{n c_{1}} \sum_{i = 1}^{n} c_{1} \log (c_{1}) = \log (n c_{1}) - \log (c_{1}) = \log (n) .

□

Theorem 3.

The polyhex nanotori T (Figure 5) is vertex-transitive.

Proof.

Suppose p and q are even. Consider two vertices u_ij and u_rs in which both integers i,r are either odd, or even, and suppose σ, π are two permutations that σ(u_it) = u_rt, 1 ≤ t ≤ p and π(u_tj) = u_ts, 1 ≤ t ≤ q. Then, σ and π are automorphisms of T in which πσ maps u_ij to u_rs and so, they are in the same orbit. Suppose now i is odd and r is even or i is even and r is odd. Then, the permutation θ which maps u_ij to u_(p+1−i)j is a graph automorphism which implies that u_ij and u_rs are in the same orbit of Aut(G) and we are done. □

Example 1.

Consider the 2-dimensional graph of zig-zag polyhex nanotori T[p,q], as depicted in Figure 6. It can be easily seen that |V(T[p,q])|=pq. By Theorem 3, T[p,q] is vertex-transitive and if for all i≠j, c_i=c_j then, by Theorem 2, we have

I f_{σ} (T [p, q]) = \log (n) = \log (p) + \log (q)

Let G be an edge-transitive graph but not vertex-transitive, and e = xy be an arbitrary edge in G. For an arbitrary edge e = uv, there exists an automorphism σ such that σ(e) = f. Hence,

{σ (u), σ (v)} = {σ (x), σ (y)} .

Let

σ (v_{i}) = σ (x)

for 1 ≤ i ≤ r and

σ (v_{j}) = σ (y)

for

r + 1 \leq j \leq n

. As a particular case, we consider c_i = c_j for all i≠j. By some elementary calculations, we obtain

I f_{σ} (G) = \log (r σ (x) + (n - r) σ (y)) - \frac{r σ (x) \log (σ (x)) + (n - r) σ (y) \log (σ (y))}{r σ (x) + (n - r) σ (y)} .

Hence, we proved the following theorem.

Theorem 4.

Let G be edge-transitive but not vertex-transitive and let c_i = c_j for all i≠j. Then,

I f_{σ} (G) = \log (r σ (x) + (n - r) σ (y)) - \frac{r σ (x) \log (σ (x)) + (n - r) σ (y) \log (σ (y))}{r σ (x) + (n - r) σ (y)} .

Example 2.

As usual, suppose that S_n indicates an star graph on n + 1 vertices. Suppose x is the central vertex and denotes the other vertices by

u_{1}, u_{2}, \dots, u_{n}

. Then d(x,u₁) = 1 and d(u_i,u_j) = 2 (1 ≤ i,j ≤ n). If c_i = c_j for all i≠j, then, by using Theorem 4, we infer that

I f_{σ} (S_{n}) = \log (2 (n - 1) + 1) - \frac{2 (n - 1) \log 2 + \log 1}{2 n - 1} = \log (2 n - 1) + \frac{1}{2 n - 1} - 1 .

Theorem 5.

Suppose G is a graph and V₁, …, V_k are all orbits of Aut(G) on V(G). Then,

I f_{σ} (G) = \log (\sum_{i = 1}^{k} σ (x_{i}) \sum_{j = 1}^{| V_{i} |} c_{j}) - \sum_{i = 1}^{k} σ (x_{i}) \sum_{j = 1}^{| V_{i} |} \frac{c_{j}}{\sum_{t = 1}^{k} σ (x_{t}) \sum_{l = 1}^{| V_{i} |} c_{l}} \log (c_{j} σ (x_{i})) .

Proof.

For all

x_{i}, x_{j} \in V_{i}

, we have

σ (x_{i}) = σ (x_{j})

and, the proof is complete. □

Suppose for the vertex v of graph G, d(v) shows the degree of vertex v. If

f (v) \in {σ (v), ​ d (v), ​ ​ d (v) σ (v), \frac{σ (v)}{​ d (v)}, \frac{d (v)}{σ (v)}}

, and we put f(v) in Equation (1), then we achieve four new entropy-based measures as follows:

I_{\deg} (G) = \log (\sum_{i = 1}^{n} d (v_{i})) - \sum_{i = 1}^{n} \frac{d (v_{i})}{\sum_{j = 1}^{n} d (v_{j})} \log (\frac{d (v_{i})}{\sum_{j = 1}^{n} d (v_{j})}) .

I_{ξ} (G) = \log (\sum_{i = 1}^{n} d (v_{i}) σ (v_{i})) - \sum_{i = 1}^{n} \frac{d (v_{i}) σ (v_{i})}{\sum_{j = 1}^{n} d (v_{j}) σ (v_{j})} \log (\frac{d (v_{i}) σ (v_{i})}{\sum_{j = 1}^{n} d (v_{j}) σ (v_{j})}) .

I_{ξ}^{D} (G) = \log (\sum_{i = 1}^{n} \frac{d (v_{i})}{σ (v_{i})}) - \sum_{i = 1}^{n} \frac{d (v_{i}) / σ (v_{i})}{\sum_{j = 1}^{n} d (v_{j}) / σ (v_{j})} \log (\frac{d (v_{i}) / σ (v_{i})}{\sum_{j = 1}^{n} d (v_{j}) / σ (v_{j})}) .

I_{ξ_{D}} (G) = \log (\sum_{i = 1}^{n} \frac{σ (v_{i})}{d (v_{i})}) - \sum_{i = 1}^{n} \frac{σ (v_{i}) / d (v_{i})}{\sum_{j = 1}^{n} σ (v_{j}) / d (v_{j})} \log (\frac{σ (v_{i}) / d (v_{i})}{\sum_{j = 1}^{n} σ (v_{j}) / d (v_{j})}) .

It is clear that since a fullerene is 3-regular, then m = 3n/2 (the number of edges) and

\begin{array}{l} I_{\deg} (G) & = \log (2 m) - \frac{1}{2 m} \sum_{i = 1}^{n} d (v_{i}) \log (d (v_{i})), \\ = \log (3 n) - \frac{1}{3 n} \sum_{i = 1}^{n} d (v_{i}) \log (d (v_{i})) . \end{array}

Moreover, the entropies of graphs based on the eccentricities of vertices are studied in [22,23,24,25,26,27,28,29,30].

4.2. Ecc-Entropy of Fullerene Graphs

In this section, similarly to the definition of the fullerene A_10n in the last section, we present two infinite families of fullerenes namely C_24n+12 and C_12n+2 with respectively 24n + 12 and 12n + 2 vertices as depicted, respectively, in Figure 7 and Figure 8. For more details about the construction of these classes of fullerenes, see references [25,27].

Theorem 6.

If c_i’s are equal, then Equation (5) yields that for n ≥ 7

\begin{array}{l} I f_{σ} (C_{24 n + 12}) & = 2 + \log 3 + \log (3 n^{2} + 12 n + 5) \\ - \frac{1}{3 n^{2} + 12 n + 5} ((n + 5) \log (n + 5) + 2 \sum_{i = 6}^{n + 5} (n + i) \log (n + i)) . \end{array}

Proof.

Consider the vertices of the central hexagon and other vertices of C_24n+12 as shown in Figure 7. Consider the eccentric contribution of each vertex as reported in Table 1. As shown in this table, there are two types of vertices. The vertices of the central and outer hexagons and the other vertices. By Equation (2) we have

\begin{array}{l} I f_{σ} (C_{24 n + 12}) & = \log (12 (n + 5) + 24 \sum_{i = 6}^{n + 5} (n + i)) \\ - \frac{1}{12 (n + 5) + 24 \sum_{i = 6}^{n + 5} (n + i)} (12 (n + 5) \log (n + 5) + 24 \sum_{i = 6}^{n + 5} (n + i) \log (n + i)) \\ = 2 + \log 3 + \log (3 n^{2} + 12 n + 5) \\ - \frac{1}{3 n^{2} + 12 n + 5} ((n + 5) \log (n + 5) + 2 \sum_{i = 6}^{n + 5} (n + i) \log (n + i)) \end{array}

□

The exceptional cases are given in Table 2.

Theorem 7.

If c_i’s are equal, then the entropy of fullerene C_12n+2, n ≥ 10 (see Figure 8) is

\begin{array}{l} I f_{σ} (C_{12 n + 2}) & = 1 + \log (n) + \log (9 n + 14) \\ - \frac{1}{n (9 n + 14)} (8 n + 11 n \log (n) + 6 \sum_{i = 1}^{n} (n + i) \log (n + i)) \end{array}

Proof.

Similarly, to the proof of Theorem 6 and by using the eccentricity of vertices as reported in Table 3 and by Equation (5), we obtain

\begin{array}{l} I f_{σ} (C_{12 n + 2}) & = \log (22 n + 12 \sum_{i = 1}^{n} (n + i)) \\ - \frac{1}{22 n + 12 \sum_{i = 1}^{n} (n + i)} (16 n \log (2 n) + 6 n \log (n) + 12 \sum_{i = 1}^{n} (n + i) \log (n + i)) \\ = 1 + \log (9 n^{2} + 14 n) - \frac{1}{n (9 n + 14)} (8 n + 11 n \log (n) + 6 \sum_{i = 1}^{n} (n + i) \log (n + i)) . \end{array}

□

The exceptional cases are given in Table 4.

4.3. Eigen—Entropy of Fullerenes

The adjacency matrix A(G) of graph G with vertex set V(G) = {v₁, v₂,..., v_n} is the n × n symmetric matrix [a_ij ] such that a_ij = 1 if v_i and v_j are adjacent and 0, otherwise. The characteristic polynomial of graph G is defined as [116]

ϕ(G,x) = det(A(G) – xI)

The roots of the characteristic polynomial are named the eigenvalues of graph G, which form the spectrum of this graph. If α is an eigenvalue of matrix A, then there exists a vector such as V, in which A.V = αV.

Let λ₁,..., λ_n be the eigenvalues of A(G); then, the energy of G is defined [117,118] as

ε (G) = \sum_{i = 1}^{n} | λ_{i} |

In theoretical chemistry, the energy is a graph parameter stemming from the Hückel molecular orbital approximation for the total π-electron energy. Thus, the graph energy has some specific chemical interests and has been extensively studied [119,120].

One information functional is based on absolute value of eigenvalues; see [121,122]. The eigen-entropy based on non-zero eigenvalues denoted by If_λ(G) is defined as follows:

I f_{λ} (G) = \log (\sum_{i = 1}^{n} c_{i} | λ_{i} |) - \sum_{i = 1}^{n} \frac{c_{i} | λ_{i} |}{\sum_{j = 1}^{n} c_{j} | λ_{j} |} \log (c_{i} | λ_{i} |)

where

λ_{i} \neq 0 (i = 1, \dots, n - 1)

. If c_is are equal, then

\begin{array}{l} I f_{λ} (G) & = \log (\sum_{i = 1}^{n} | λ_{i} |) - \sum_{i = 1}^{n} \frac{| λ_{i} |}{\sum_{j = 1}^{n} | λ_{j} |} \log (| λ_{i} |) \\ = \log (ε (G)) - \frac{1}{ε (G)} \sum_{i = 1}^{n} | λ_{i} | \log (| λ_{i} |) . \end{array}

(6)

Theorem 8.

Let G be a connected graph with non-zero eigenvalues

λ_{i} \neq 0 (i = 1, \dots, n - 1)

. Then

\log (2 \sqrt{m})) - \frac{2 m}{ε (G)} \leq I_{λ} (G) \leq \log (2 m)) - \frac{α^{2} s}{ε (G)},

where

α = \min {| λ_{i} | : i = 1, \dots, s}

and where s is the number of distinct eigenvalues.

Proof.

It is a well-known fact that

ε (G) \leq 2 m

. By using Equation (6), we have

I_{λ} (G) \leq \log (2 m) - \frac{1}{2 m} (\sum_{i = 1}^{s} | λ_{i} | \log (| λ_{i} |)) .

Since logarithm is an increasing function, we have

\log | λ_{i} | \leq | λ_{i} |

. On the other hand,

\sum_{i = 1}^{s} λ_{i}^{2} = 2 m

yields

\log (2 \sqrt{m})) - \frac{2 m}{ε (G)} \leq I_{λ} (G) \leq \log (2 m)) - \frac{α^{2} s}{ε (G)}

Now,

2 \sqrt{m} \leq ε (G) \leq 2 m

implies that

\log (2 \sqrt{m})) - \frac{2 m}{ε (G)} \leq I_{λ} (G) \leq \log (2 m)) - \frac{\sum_{i = 1}^{s} {(\log | λ_{i} |)}^{2}}{ε (G)}

(7)

Let

α = \min {| λ_{i} | : i = 1, \dots, s}

. The nullity of G is 1. By reformulating Equation (7), one can see that

\log (2 \sqrt{m})) - \frac{2 m}{ε (G)} \leq I_{λ} (G) \leq \log (2 m)) - \frac{α^{2} s}{ε (G)} .

□

Theorem 9.

Let G be a graph whose eigenvalues are in the interval [−1,1]. Then

I_{λ} (G) \geq ε (G)

Proof.

Clearly, we obtain

\begin{array}{l} I_{λ} (G) & = \log (ε (G)) - \frac{1}{ε (G)} \sum_{i = 1, λ \neq 0}^{n} | λ_{i} | \log | λ_{i} | \\ = \log (ε (G)) - \frac{1}{ε (G)} {\sum_{i = 1, λ \neq 0}^{n} \log | λ_{i} |}^{| λ_{i} |} \\ = \log (ε (G)) - \frac{1}{ε (G)} \log (Π | λ_{i} |^{^{| λ_{i} |}}) . \end{array}

On the other hand,

Π | λ_{i} |^{| λ_{i} |} \leq Π λ_{1}^{| λ_{i} |} = λ_{1}^{ε (G)}

. So

I_{λ} (G) \geq \log (ε (G)) - \log (λ_{1}) = \log (\frac{ε (G)}{λ_{1}}),

Since

λ_{i} \in [- 1, 1]

, thus

I_{λ} (G) \geq ε (G)

. □

Theorem 10.

Let G be a graph and m_∆(G) be the ∆-th spectral moment of G. Then

I_{λ} (G) \geq ε (G) - \frac{m_{Δ} (G)}{ε (G)} .

Proof.

We have

\sum_{i = 1}^{n} | λ_{i} | \log | λ_{i} | \leq {\sum_{i = 1}^{n} \log | λ_{i} |}^{| λ_{i} |} \leq \sum_{i = 1}^{n} \log | λ_{i} |^{Δ} \leq \sum_{i = 1}^{n} | λ_{i} |^{Δ} = m_{Δ} (G) .

Thus

I_{λ} (G) \geq ε (G) - \frac{m_{Δ} (G)}{ε (G)} . □

4.4. The Hosoya Entropy of Fullerenes

Here, we introduce the Hosoya entropy based on the distance between vertices of a graph, see [123,124,125]. Given a graph G and two vertices u and v, the distance between them is defined as the length of the shortest path connecting them. Let s_i(u) be the number of vertices at distance i from vertex u. Then the sequence dds(v) = (s₀(v), s₁(v),…, s_d(v)) is called the distance degree sequence of v.

Two vertices u and v are H-equivalent if dds(u) = dds(v) see [126] and the class of H -equivalent vertices constitutes the H-partitions of a graph.

Suppose G has h H-equivalent classes and the cardinality of ith H-equivalent class is n_i (1 ≤ i ≤ h). Then, H-entropy of G [127] is

H (G) = - \sum_{i = 1}^{h} h_{i} \log (h_{i})

where

h_{i} = n_{i} / | V |

Let also O₁,…, O_l be all the orbits of Aut(G) in the set of vertices. If l_i is the cardinality of the i-th orbit for 1 ≤ i ≤ l, then the orbit entropy of G [116,117,118,119] is given by

I_{a} (G) = - \sum_{i = 1}^{l} μ_{i} \log (μ_{i})

where

μ_{i} = | n_{i} | / | V |

The value of H(G) for a graph G with n vertices is between zero and log(n). The minimum is achieved when all the vertices are H-equivalent; the maximum is reached when a non-pair of vertices are H-equivalent. Complete graphs and cycles have a H-entropy of 0.

Remark 1.

In a vertex-transitive graph, all the vertices are H-equivalent and the H-entropy is zero. The converse of this fact is not true. In other words, there are many examples of non-vertex-transitive graphs with zero H-entropy.

Theorem 11.

Two vertices in the same orbit are H-equivalent. In addition, the H-entropy of a vertex-transitive graph is zero [128].

Although two similar vertices are H-equivalent, the converse is not true. For example, in Figure 9, u and v are H-equivalent, and thus D(u) = D(v), while they are in different orbits. Moreover, two vertices u and v in Figure 10 are not H-equivalent but D(u) = D(v).

Theorem 12

([128]). If G is a regular graph with diameter ρ = 2, then H(G) = 0.

Theorem 13.

If G is a regular graph with non-zero H-entropy, then its diameter is greater than 2.

Theorem 14

([128]). If G is a connected graph on n vertices, then H(G) = n if and only if Aut(G) is trivial.

Theorem 15

([128]). The H-entropy of a regular graph of degree greater than n/2 is zero.

Theorem 16

([128]). The H-entropy of a regular graph of degree greater than or equal with n/2 is zero, if the number of vertices is an even number and G is edge-transitive.

Theorem 17

([128]). If G is a graph on at least five vertices which is edge-transitive but not bipartite, and each vertex has odd degree. Then, H-entropy is zero.

In continuing, we compute the H-entropy of graphs with at most two orbits.

Definition 1.

A graph G is called co-distance, if for each pair of vertices (u,v) their total distances are the same, namely D(u) = D(v).

Each vertex-transitive graph is co-distance, but there are examples of co-distance non-transitive graphs. The graph G in Figure 11 has two orbits V₁ = {1, 2, 5, 6, 8, 12} and V₂ = {3, 10, 4, 7, 11, 9}. It is not difficult to see that d(G) = 4 and D(1) = D(3) while the H-entropy is not zero.

Theorem 18

([128]). Up to isomorphism there are exactly 14 co-distance regular graphs of order at most 14 with two orbits and diameter greater than or equal with 3. Among them, only the graph G depicted in Figure 11 has non-zero H-entropy.

Theorem 19

([128]). Suppose the H-entropy of graph G is zero. Then nor G is a tree or a regular graph.

Theorem 20

([128]). Suppose G is a graph with two orbits V₁ and V₂ and non-zero H-entropy. Then H(G) = 1 if and only if n is even and |V₁| = |V₂| = n/2.

Theorem 21

([128]). Let G be a regular graph with two orbits and diameter less than four. If G is co-distance, then its H-entropy is zero.

In the final part of this section, we compute the H-entropy of some infinite classes of fullerene graphs.

Theorem 22.

The fullerene graph A_{12n + 4} where n ≥ 4 satisfies

H (A_{12 n + 4}) = \log (12 n + 4) - \frac{1}{12 n + 4} ((12 n + 3) \log 3 + 6 (2 n - 4)) .

Proof.

In Figure 12, the vertices of ith (i = 1, 2, 3, 4, 5) layer are in the same orbit. The vertices of the layers 6, 7,…, n constitute three orbits labeled by the numbers 1, 2, and 3. Vertices with labels 2, 3 are in the same H-equivalent partition and the other vertices compose the H-equivalent partition. Finally, the vertices of the outer pentagon are H-equivalent. In other words, there are an equivalence class of size 1, nine equivalence classes of size 3, and 2n − 4 equivalence classes of size 6 which yield

\begin{array}{l} H (A_{12 n + 4}) & = \frac{1}{12 n + 4} \log (12 n + 4) + \frac{27}{12 n + 4} \log (\frac{12 n + 4}{3}) \\ + \frac{6 (2 n - 4)}{12 n + 4} \log (\frac{12 n + 4}{6}) \\ = \log (12 n + 4) - (\frac{27}{12 n + 4} \log 3 + \frac{6 (2 n - 4)}{12 n + 4} (1 + \log 3)) \\ = \log (12 n + 4) - \frac{1}{12 n + 4} ((12 n - 24) + (12 n + 3) \log 3) . \end{array}

□

Similarly to the structure of A_10n in last section, the fullerene graph A_12n+4 is composed of a nanotube Tz[6, n − 10] together with two caps B₁ and B₂, see Figure 13, Figure 14, Figure 15 and Figure 16. Thus, the vertices of A_12n+4 are labeled as given in Figure 17. The H-partitions and the eccentricity of vertices of caps B₁ and B₂ are given in Table 5.

4.5. Radial Entropy and Orbit Measures

In a network, knowing, for example, the degree of vertices gives us essential information about the number of interconnections of each component. These data provide a narrow understanding of complex networks, because the vertex partition [12] based on degree is coarser than automorphism partition. Hence, it is important to know if we substitute the degree partitions instead of orbit partitions; then, the study of complex network in the view of symmetry leads us to understand more about the structure of the regarding graph. The orbit entropy is defined as

I_{a} (G) = - \sum_{i = 1}^{k} \frac{| O_{i} |}{| V |} \log (\frac{| O_{i} |}{| V |})

where O_i’s (1 ≤ i ≤ k) are orbits of G. Regarding the orbit polynomial, the symmetry index S(G) is defined [127] as follows:

S (G) = \log (n) - I_{a} (G) + \log (| Aut (G) |) .

Suppose

N_{i_{j}}

, 1 ≤ j ≤ k is the collection of all vertices with eccentricity i_j. Then, the radial entropy is defined by

H_{e c c} (G) = - \sum_{i = 1}^{k} \frac{| N_{i} |}{| V |} \log (\frac{| N_{i} |}{| V |}) .

(8)

Let p₁,..., p_n−₁₀ be the H-equivalent classes of Tz[6,n − 10] which contains the vertices with label i. Then ecc(p_i) = 2n − i-9. Thus, the eccentricity sequence of fullerene graph A_12n+4 is

{{(2 n - i)}^{12} (1 \leq i \leq n - 1), {(2 n)}^{9}, {(2 n + 1)}^{7}} .

(9)

Theorem 23

([129]). Consider the fullerene graph A_12n+4, where n ≥ 4. If n is even, then

I_{a} (A_{12 n + 4}) = \log (12 n + 4) - \frac{1}{12 n + 4} ((12 n + 3) \log 3 + 6 n) .

If n is odd, then

I_{a} (A_{12 n + 4}) = \log (12 n + 4) - \frac{1}{12 n + 4} ((12 n + 3) \log 3 + 6 n + 6) .

Theorem 24

([129]). The radial entropy of fullerene A12n + 4 (n ≥ 4) is

\begin{array}{l} H_{e c c} (A_{12 n + 4}) & = \log (12 n + 4) \\ - \frac{1}{12 n + 4} (24 (n - 1) + (12 n + 6) \log 3 + 7 \log 7) . \end{array}

Theorem 25

([129]). If the ci’s are equal, the entropy of fullerene A12n + 4 (n ≥ 11) is given by

\begin{array}{l} I f_{σ} (A_{12 n + 4}) & = \log (18 n^{2} + 14 n + 7) - \frac{1}{18 n^{2} + 14 n + 7} ((14 n + 7) \log (2 n + 1)) \\ + 18 n \log (2 n) + 12 A, \end{array}

where

A = \sum_{i = 1}^{n - 1} (2 n - i) \log (2 n - i) .

Theorem 26

([129]). The degree-based entropy of fullerene graph A12n + 4 is

D (A_{12 n + 4}) = \log (36 n + 12) - \frac{(36 n + 12) \log 3}{36 n + 12} = \log (12 n + 4) .

5. Correlation Analysis

The Pearson correlations between all entropies introduced in this paper are given in Equation (1), for fullerene graph A_12n+4.

\begin{array}{l} \begin{matrix} I f_{σ} & H & D & I_{a} & H_{e c c} \end{matrix} \\ \begin{matrix} I f_{σ} \\ H \\ D \\ I_{a} \\ H_{e c c} \end{matrix} (\begin{matrix} 0.999871662 & 1 & 0.99999826 & 0.999988517 \\ 0.999872017 & 0.999899809 & 0.999936954 \\ 0.999998301 & 0.999988624 \\ 0.999995717 \end{matrix}) \end{array}

(10)

Moreover, the exact values of energy and five entropies for fullerene graph A_12n+4 (11 ≤ n ≤ 20) is reported in Table 6. These results show that the correlation between energy and each entropy measure of A_12n+4 is greater than 0.99 (see Table 7). It can be found that they release the same structural information about regarding fullerene. In [129], the authors found a similar result for a different class of fullerenes.

Author Contributions

Conceptualization, formal analysis, project administration M.G. and M.D.; methodology, software, F.E.-S. All authors have read and agreed to the published version of the manuscript.

Funding

Modjtaba Ghorbani is partially supported by the Shahid Rajaee Teacher Training University. Also, Matthias Dehmer thanks the Austrian Science Funds for supporting this work (project P30031).

Acknowledgments

This research is partially supported by Shahid Rajaee Teacher Training University. Matthias Dehmer thanks the Austrian Science Fund for supporting this work (P 30031).

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Illustration of two cubic graphs: (a) Cuneane; (b) cube.

Figure 2. The 2-D graph of zig-zag nanotube T_Z[5,10].

Figure 3. Cap B.

Figure 4. The presentation of fullerene A_10n as a combining of two copies of B and the nanotube T_Z[5,n − 4].

Figure 5. Three-dimensional graph of zig-zag polyhex nanotori.

Figure 6. Two-dimensional lattice for T[p,q].

Figure 7. The molecular graph of the fullerene C_24n+12, for n = 3.

Figure 8. The molecular graph of the fullerene C_12n+2.

Figure 9. Two vertices with the same total distance which are not H-equivalent.

Figure 10. The vertices u and v are not H-equivalent but D(u) = D(v).

Figure 11. A cubic co-distance graph of diameter 4 with non-zero H-entropy.

Figure 12. The fullerene A_12n+4.

Figure 13. The orbits of the ith layer (2 ≤ i ≤ n)of the fullerene graph A_12n+4.

Figure 14. The subgraph B₁.

Figure 15. The subgraph B₂.

Figure 16. The 3-dimensional structure of fullerene graph A_12n+4.

Figure 17. The Hosoya-partitions of Tz[6,n − 10].

Table 1. The eccentricity of vertices of C_24n+12, n ≥ 7.

Vertices	σ (x)	No
Vertices of type 1	n + 5	12
Vertices of type 2	n + i (6 ≤ i ≤ n + 5)	24

Table 2. Some exceptional cases of C_24n+12 fullerenes.

Fullerenes	If_σ(F)
C₆₀	log 60
C₈₄	log 84
C₁₀₈	$\log 1320 - \frac{1008 \log 12 + 312 \log 13}{1320}$
C₁₃₂	$\log 1728 - (\frac{720 \log 12 + 312 \log 13 + 336 \log 14 + 360 \log 15}{1728})$
C₁₅₆	$\log 2232 - (\frac{432 \log 12 + 312 \log 13 + 336 \log 14 + 360 \log 15 + 384 \log 16 + 408 \log 17}{2232})$

Table 3. The eccentricity of vertices of C_12n+2, n ≥ 10.

Vertices	σ(x)	No.
Vertices of type 1	2n	8
Vertices of type 2	n	6
Other Vertices	n + i (1 ≤ i ≤ n)	12

Table 4. Some exceptional cases of C_12n+2 fullerenes.

F	If_σ(F)
C₂₆	$\log 132 - \frac{120 \log 5 + 12 \log 6}{132}$
C₃₈	$\log 266 - \log 7 = \log 38$
C₅₀	$\log 392 - \frac{84 \log 7 + 272 \log 8 + 36 \log 9}{392}$
C₆₂	$\log 548 - \frac{192 \log 8 + 216 \log 9 + 140 \log 10}{548}$
C₇₄	$\log 720 - \frac{96 \log 8 + 216 \log 9 + 180 \log 10 + 132 \log 11 + 96 \log 12}{720}$
C₈₆	$\log 940 - \frac{216 \log 9 + 180 \log 10 + 132 \log 11 + 144 \log 12 + 156 \log 13 + 112 \log 14}{940}$
C₉₈	$\log 1088 - \frac{108 \log 9 + 180 \log 10 + 132 \log 11 + 144 \log 12 + 156 \log 13 + 168 \log 14 + 180 \log 15 + 128 \log 16}{1088}$
C₁₁₀	$\log 1500 - \frac{180 \log 10 + 132 \log 11 + 144 \log 12 + 156 \log 13 + 168 \log 14 + 180 \log 15 + 128 \log 16 + 204 \log 17 + 144 \log 18}{1088}$

Table 5. The H-partition and eccentricity of fullerene graph.

Partitions	Elements	ecc
V₁	1	2n + 1
V_{2n + 6}	12n − 1, 12n, 12n + 1, 12n + 2, 12n + 3, 12n + 4
V₂	2, 5, 8	2n
V_{2n + 5}	12n − 13, 12n − 11, 12n – 9, 12n − 7, 12n − 5, 12n − 3
V₃	3, 4, 6, 7, 9, 10	2n − 1
V_{2n + 4}	12n − 12, 12n − 10, 12n – 8, 12n − 6, 12n − 4, 12n − 2
V₄	12, 14, 16, 18, 20, 22	2n − 2
V_{2n + 3}	12n − 25, 12n − 23, 12n − 21, 12n − 19, 12n − 17, 12n − 15
V₅	11, 15, 19	2n − 3
V₆	13, 17, 21
V_{2n + 2}	12n − 24, 12n − 22, 12n − 20, 12n − 18, 12n − 16, 12n − 14
V₇	23, 27, 31	2n − 4
V₈	25, 29, 33
V_{2n + 1}	12n − 36, 12n − 34, 12n − 32, 12n − 30, 12n − 28, 12n − 26
V₉	24, 26, 28, 30, 32, 34	2n − 5
V_2n	12n − 37, 12n − 35, 12n − 33, 12n − 31, 12n − 29, 12n − 27
V₁₀	36, 38, 40, 42, 44, 46	2n − 6
V_2n−1	12n − 49, 12n − 47, 12n − 45, 12n − 43, 12n − 41, 12n − 39
V₁₁	35, 39, 43	2n − 7
V₁₂	37, 41, 45
V_{2n − 2}	12n − 48, 12n − 46, 12n − 44, 12n − 42, 12n − 40, 12n − 38
V₁₃	47, 51, 55	2n − 8
V₁₄	49, 53, 57
V_{2n − 3}	12n − 60, 12n − 58, 12n − 56, 12n − 54, 12n − 52, 12n − 50
V₁₅	48, 50, 52, 54, 56, 58	2n − 9
V_{2n − 4}	12n − 61, 12n − 59, 12n − 57, 12n − 55, 12n − 55, 12n − 53, 12n − 51

Table 6. The graph energy and five types of entropies applied to A_12n+4.

n	E	D	If_σ	I_a	H	H_ecc
11	212.87	7.08	7.06	5.02	4.72	3.57
12	231.73	7.2	7.18	5.14	4.82	3.68
13	250.59	7.32	7.29	5.25	4.92	3.79
14	269.46	7.42	7.39	5.36	5.01	3.89
15	288.32	7.52	7.49	5.45	5.09	3.98
16	307.19	7.61	7.58	5.54	5.18	4.07
17	326.05	7.7	7.67	5.63	5.25	4.15
18	344.91	7.78	7.75	5.71	5.33	4.23
19	363.78	7.85	7.85	5.78	5.4	4.31
20	382.64	7.93	7.9	5.86	5.46	4.38

Table 7. The correlation between graph energy and entropies applied to A_12n+4.

	E,D	E, If_σ	E,I_a	E,H	E,H_ecc
Cor	0.9964006	0.9972326	0.99673	0.9975728	0.9974525

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Ghorbani, M.; Dehmer, M.; Emmert-Streib, F. Properties of Entropy-Based Topological Measures of Fullerenes. Mathematics 2020, 8, 740. https://doi.org/10.3390/math8050740

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Ghorbani M, Dehmer M, Emmert-Streib F. Properties of Entropy-Based Topological Measures of Fullerenes. Mathematics. 2020; 8(5):740. https://doi.org/10.3390/math8050740

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Ghorbani, Modjtaba, Matthias Dehmer, and Frank Emmert-Streib. 2020. "Properties of Entropy-Based Topological Measures of Fullerenes" Mathematics 8, no. 5: 740. https://doi.org/10.3390/math8050740

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Ghorbani, M., Dehmer, M., & Emmert-Streib, F. (2020). Properties of Entropy-Based Topological Measures of Fullerenes. Mathematics, 8(5), 740. https://doi.org/10.3390/math8050740

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Properties of Entropy-Based Topological Measures of Fullerenes

Abstract

1. Introduction

2. Definitions and Preliminaries

3. Fullerene Graphs

4. Entropy Measure

4.1. Eccentric Entropy Measure

4.2. Ecc-Entropy of Fullerene Graphs

4.3. Eigen—Entropy of Fullerenes

4.4. The Hosoya Entropy of Fullerenes

4.5. Radial Entropy and Orbit Measures

5. Correlation Analysis

Author Contributions

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Acknowledgments

Conflicts of Interest

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