# Properties of Entropy-Based Topological Measures of Fullerenes

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{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).

## 1. Introduction

## 2. Definitions and Preliminaries

_{1}, V

_{2},..., 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.

_{n}; then, the corresponding permutation is $\rho =(1,2,\dots ,n)$ which yields that C

_{n}is vertex-transitive.

_{3}, 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.

## 3. Fullerene Graphs

_{60}which contains 60 carbon atoms, 12 pentagons and 20 hexagons, is vertex-transitive.

_{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

_{1},…,p

_{n}) that satisfies in two conditions 0 ≤ p

_{i}≤ 1 and ${\sum}_{i=1}^{n}{p}_{i}}=1$. The Shannon’s entropy is $I(p)=-{\displaystyle {\sum}_{i=1}^{n}{p}_{i}\mathrm{log}{p}_{i}}$, where the symbol “log” is the logarithm on the basis 2. Let

_{i}of graph G, if we put

_{1},…,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].

#### 4.1. Eccentric Entropy Measure

_{i}):= c

_{i}σ(v

_{i}), where c

_{i}> 0 for 1 ≤ i ≤ n. Then

_{i}s are equal, then

**Theorem**

**2.**

_{i}= c

_{j}for all, i ≠ j then If

_{σ}(G) = log(n).

**Proof.**

_{i}= c

_{j}for all i ≠ j then we have

**Proof.**

_{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.**

_{i}=c

_{j}then, by Theorem 2, we have$I{f}_{\sigma}(T\left[p,q\right])=\mathrm{log}\left(n\right)=\mathrm{log}\left(p\right)+\mathrm{log}(q)$.

_{i}= c

_{j}for all i≠j. By some elementary calculations, we obtain

**Theorem**

**4.**

_{i}= c

_{j}for all i≠j. Then,

**Example**

**2.**

_{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,u

_{1}) = 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

**Theorem**

**5.**

_{1}, …, V

_{k}are all orbits of Aut(G) on V(G). Then,

**Proof.**

#### 4.2. Ecc-Entropy of Fullerene Graphs

_{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.**

_{i}’s are equal, then Equation (5) yields that for n ≥ 7

**Proof.**

_{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

**Theorem**

**7.**

_{i}’s are equal, then the entropy of fullerene C

_{12n+2}, n ≥ 10 (see Figure 8) is

**Proof.**

#### 4.3. Eigen—Entropy of Fullerenes

_{1}, v

_{2},..., 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]

_{1},..., λ

_{n}be the eigenvalues of A(G); then, the energy of G is defined [117,118] as

_{λ}(G) is defined as follows:

_{i}s are equal, then

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

_{∆}(G) be the ∆-th spectral moment of G. Then

**Proof.**

#### 4.4. The Hosoya Entropy of Fullerenes

_{i}(u) be the number of vertices at distance i from vertex u. Then the sequence dds(v) = (s

_{0}(v), s

_{1}(v),…, s

_{d}(v)) is called the distance degree sequence of v.

_{i}(1 ≤ i ≤ h). Then, H-entropy of G [127] is

_{1},…, 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

**Remark**

**1.**

**Theorem**

**11.**

**Theorem**

**13.**

**Theorem**

**14**

**Theorem**

**16**

**.**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**

**.**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.

**Definition**

**1.**

_{1}= {1, 2, 5, 6, 8, 12} and V

_{2}= {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**

**Theorem**

**19**

**Theorem**

**20**

**.**Suppose G is a graph with two orbits V

_{1}and V

_{2}and non-zero H-entropy. Then H(G) = 1 if and only if n is even and |V

_{1}| = |V

_{2}| = n/2.

**Theorem**

**21**

**.**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.

**Theorem**

**22.**

_{12n + 4}where n ≥ 4 satisfies

**Proof.**

_{10n}in last section, the fullerene graph A

_{12n+4}is composed of a nanotube Tz[6, n − 10] together with two caps B

_{1}and B

_{2}, 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 B

_{1}and B

_{2}are given in Table 5.

#### 4.5. Radial Entropy and Orbit Measures

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

_{j}. Then, the radial entropy is defined by

_{1},..., 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

**Theorem**

**23**

**.**Consider the fullerene graph A

_{12n+4}, where n ≥ 4. If n is even, then

**Theorem**

**24**

**.**The radial entropy of fullerene A12n + 4 (n ≥ 4) is

**Theorem**

**25**

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

**Theorem**

**26**

**.**The degree-based entropy of fullerene graph A12n + 4 is

## 5. Correlation Analysis

_{12n+4}.

_{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

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**The presentation of fullerene A

_{10n}as a combining of two copies of B and the nanotube T

_{Z}[5,n − 4].

Vertices | σ (x) | No |
---|---|---|

Vertices of type 1 | n + 5 | 12 |

Vertices of type 2 | n + i (6 ≤ i ≤ n + 5) | 24 |

Fullerenes | If_{σ}(F) |
---|---|

C_{60} | log 60 |

C_{84} | log 84 |

C_{108} | $\mathrm{log}1320-\frac{1008\mathrm{log}12+312\mathrm{log}13}{1320}$ |

C_{132} | $\mathrm{log}1728-\left(\frac{720\mathrm{log}12+312\mathrm{log}13+336\mathrm{log}14+360\mathrm{log}15}{1728}\right)$ |

C_{156} | $\mathrm{log}2232-\left(\frac{432\mathrm{log}12+312\mathrm{log}13+336\mathrm{log}14+360\mathrm{log}15+384\mathrm{log}16+408\mathrm{log}17}{2232}\right)$ |

Vertices | σ(x) | No. |
---|---|---|

Vertices of type 1 | 2n | 8 |

Vertices of type 2 | n | 6 |

Other Vertices | n + i (1 ≤ i ≤ n) | 12 |

F | If_{σ}(F) |
---|---|

C_{26} | $\mathrm{log}132-\frac{120\mathrm{log}5+12\mathrm{log}6}{132}$ |

C_{38} | $\mathrm{log}266-\mathrm{log}7=\mathrm{log}38$ |

C_{50} | $\mathrm{log}392-\frac{84\mathrm{log}7+272\mathrm{log}8+36\mathrm{log}9}{392}$ |

C_{62} | $\mathrm{log}548-\frac{192\mathrm{log}8+216\mathrm{log}9+140\mathrm{log}10}{548}$ |

C_{74} | $\mathrm{log}720-\frac{96\mathrm{log}8+216\mathrm{log}9+180\mathrm{log}10+132\mathrm{log}11+96\mathrm{log}12}{720}$ |

C_{86} | $\mathrm{log}940-\frac{216\mathrm{log}9+180\mathrm{log}10+132\mathrm{log}11+144\mathrm{log}12+156\mathrm{log}13+112\mathrm{log}14}{940}$ |

C_{98} | $\mathrm{log}1088-\frac{108\mathrm{log}9+180\mathrm{log}10+132\mathrm{log}11+144\mathrm{log}12+156\mathrm{log}13+168\mathrm{log}14+180\mathrm{log}15+128\mathrm{log}16}{1088}$ |

C_{110} | $\mathrm{log}1500-\frac{180\mathrm{log}10+132\mathrm{log}11+144\mathrm{log}12+156\mathrm{log}13+168\mathrm{log}14+180\mathrm{log}15+128\mathrm{log}16+204\mathrm{log}17+144\mathrm{log}18}{1088}$ |

Partitions | Elements | ecc |
---|---|---|

V_{1} | 1 | 2n + 1 |

V_{2n + 6} | 12n − 1, 12n, 12n + 1, 12n + 2, 12n + 3, 12n + 4 | |

V_{2} | 2, 5, 8 | 2n |

V_{2n + 5} | 12n − 13, 12n − 11, 12n – 9, 12n − 7, 12n − 5, 12n − 3 | |

V_{3} | 3, 4, 6, 7, 9, 10 | 2n − 1 |

V_{2n + 4} | 12n − 12, 12n − 10, 12n – 8, 12n − 6, 12n − 4, 12n − 2 | |

V_{4} | 12, 14, 16, 18, 20, 22 | 2n − 2 |

V_{2n + 3} | 12n − 25, 12n − 23, 12n − 21, 12n − 19, 12n − 17, 12n − 15 | |

V_{5} | 11, 15, 19 | 2n − 3 |

V_{6} | 13, 17, 21 | |

V_{2n + 2} | 12n − 24, 12n − 22, 12n − 20, 12n − 18, 12n − 16, 12n − 14 | |

V_{7} | 23, 27, 31 | 2n − 4 |

V_{8} | 25, 29, 33 | |

V_{2n + 1} | 12n − 36, 12n − 34, 12n − 32, 12n − 30, 12n − 28, 12n − 26 | |

V_{9} | 24, 26, 28, 30, 32, 34 | 2n − 5 |

V_{2n} | 12n − 37, 12n − 35, 12n − 33, 12n − 31, 12n − 29, 12n − 27 | |

V_{10} | 36, 38, 40, 42, 44, 46 | 2n − 6 |

V_{2n−1} | 12n − 49, 12n − 47, 12n − 45, 12n − 43, 12n − 41, 12n − 39 | |

V_{11} | 35, 39, 43 | 2n − 7 |

V_{12} | 37, 41, 45 | |

V_{2n − 2} | 12n − 48, 12n − 46, 12n − 44, 12n − 42, 12n − 40, 12n − 38 | |

V_{13} | 47, 51, 55 | 2n − 8 |

V_{14} | 49, 53, 57 | |

V_{2n − 3} | 12n − 60, 12n − 58, 12n − 56, 12n − 54, 12n − 52, 12n − 50 | |

V_{15} | 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 |

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 |

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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**MDPI and ACS Style**

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

**AMA Style**

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

**Chicago/Turabian Style**

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