Some Topological Invariants of Graphs Associated with the Group of Symmetries

Wei, Chang-Cheng; Salman, Muhammad; Ali, Usman; Rehman, Masood Ur; Ahmad Khan, Muhammad Aqeel; Chaudary, Muhammad Hasanain; Ahmad, Farooq

doi:https://doi.org/10.1155/2020/6289518

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Volume 2020 | Article ID 6289518 | https://doi.org/10.1155/2020/6289518

Some Topological Invariants of Graphs Associated with the Group of Symmetries

Chang-Cheng Wei,¹Muhammad Salman,²Usman Ali,^3,4Masood Ur Rehman ,^5,6Muhammad Aqeel Ahmad Khan,⁷Muhammad Hasanain Chaudary,⁸and Farooq Ahmad⁸

Academic Editor: Teodorico C. Ramalho

Received21 Aug 2019

Accepted09 Oct 2019

Published09 Mar 2020

Abstract

A topological index is a quantity that is somehow calculated from a graph (molecular structure), which reflects relevant structural features of the underlying molecule. It is, in fact, a numerical value associated with the chemical constitution for the correlation of chemical structures with various physical properties, chemical reactivity, or biological activity. A large number of properties like physicochemical properties, thermodynamic properties, chemical activity, and biological activity can be determined with the help of various topological indices such as atom-bond connectivity indices, Randić index, and geometric arithmetic indices. In this paper, we investigate topological properties of two graphs (commuting and noncommuting) associated with an algebraic structure by determining their Randić index, geometric arithmetic indices, atomic bond connectivity indices, harmonic index, Wiener index, reciprocal complementary Wiener index, Schultz molecular topological index, and Harary index.

1. Introduction

In quantitative structure-activity relationship (QSAR)/quantitative structure-property relationship (QSPR) study, physicochemical properties and topological indices such as Randić index, atom-bond connectivity (ABC) index, and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. A topological index is actually designed by transforming a chemical structure into a numerical number. It correlates certain physicochemical properties such as boiling point, stability, and strain energy of chemical compounds of a molecular structure (graph). It is a numeric quantity associated with a chemical structure (graph), which characterizes the topology of the structure and is invariant under a structure-preserving mapping [1]. In 1947, Wiener [2] introduced the concept of (distance-based) topological index while working on the boiling point of paraffin. He named this index as the path number. Later on, the path number was renamed as the Wiener index [2], and then, the theory of topological indices started. Nowadays, a number of distance-based and degree-based topological indices have been introduced and computed (see for example [3–15], and the references therein).

We consider simple and connected graph (chemical structure) G with vertex set and edge set . We denote the two adjacent vertices u and in G as and nonadjacent vertices as The number denotes the degree of a vertex and is the degrees sum of , where is the neighborhood of . The number denotes the length of a geodesic between u and in and is called the distance between u and The eccentricity of a vertex in G, denoted by , is the maximum distance between and any other vertex of G. The minimum eccentricity amongst the vertices of G is called the radius of G, denoted by . The diameter of a graph G is the maximum eccentricity in G, denoted by . A vertex in G is said to be a central vertex if , and the subgraph of G induced by central vertices of G is called the center of G. A vertex in a graph G is called a peripheral vertex if , and the subgraph of G induced by peripheral vertices is called the periphery of G. The sum of two graphs and , denoted by , is a graph with vertex set and an edge set .

Let be a group. The set is called the center of the group Then, commuting and noncommuting graphs of are defined as follows:(i)The commuting graph of a nonabelian group is denoted by with vertex set . For two distinct elements , in (x and y form an edge in ) if and only if in . The concept of commuting graphs on noncentral elements of a group has been studied by various researchers (see [16, 17]).(ii)The noncommuting graph of a nonabelian group is a graph with vertex set , and two distinct vertices u and in form an edge if in . The study of noncommuting graphs of groups was initiated in 1975 by Neumann [18] who posed the problem regarding the clique number of a noncommuting graph. Noncommuting graphs on noncentral elements of a group have also been studied by various other researchers [19, 20].

The following useful property for a noncommuting graph was proposed in [19].

Proposition 1 (see [19]). For any nonabelian group , .

A graph G is said to be self-centered if . Since for every , so we have the following straightforward proposition:

Proposition 2. A noncommuting graph of any nonabelian finite group is self-centered if for each , . Otherwise, it is the sum of the center and the periphery of .

This paper aimed at investigating all the topological properties (listed in Table 1) of commuting and noncommuting graphs associated with the group of symmetries. The rest of the paper consists of five sections. In the next section, we illustrate the group of symmetries and associate commuting and noncommuting graphs to this group. In Section 3, some useful constructions to investigate our main results of Sections 4 and 5 are provided.

2. Group of Symmetries and Associated Graphs

Group of symmetries finds its remarkable use in the theory of electron structures and molecular vibrations. Due to their significant employment in chemical structures, in the context of topological indices, we consider the group of symmetries of a regular polygon (also called a regular n-gon for ) in this paper. A regular n-gon is a geometrical figure all of whose sides have the same length and all the angles are of equal measurement. Each internal angle of a n-gon is of radian. The group of symmetries of a n-gon consists of elements, which are n rotations ( about its center through an angle of radian, where , either all clockwise or all anticlockwise) and n reflections (for even , the reflections through a line joining the midpoints of the opposite sides or through a line joining two opposite vertices, and for odd , the reflections through those lines which join a vertex with the midpoint of the opposite side). The group of symmetries is denoted by and is called the dihedral group of order . If we denote a rotation by “a” and a reflection by “b,” then elements of are and , where e is the identical rotation. The general representation of is given bywith the center

Let , and . Then and

In the case of even value of , we partitioned into two element subsets , , so that .

Remark 1. In the dihedral group , we have(i) for all (ii) for (iii)For odd values of , for distinct (iv)For even values of , and for any distinct , if and only if , (v)For any distinct , (vi)For each pair , According to Remark 1, the commuting graph on is defined in the following result.

Proposition 3 (see [16]). For all , let be a commuting graph on , then

Here, is the trivial graph, is a complete graph on p vertices, is a null (empty) graph on t vertices, and is the union of copies of .

Let , and be the corresponding noncommuting graph. Then, according to Remark 1, we have the following points:

When is odd, then(1)For whenever .(2)For whenever .(3)For whenever and .(4)In , it can be seen that for all , and for all . It follows that induces the center of , which is a complete graph on vertices, and induces the periphery of , which is a null graph on vertices.

When is even, then(1)For whenever for any .(2)For whenever .(3)For whenever and .(4)For whenever and with and .(5)In , it can be seen that for all . It follows that is a self-centered graph, which is a complete multipartite graph with partite sets , and one partite set .

Hence, by Proposition 2, we deduce the following result.

Proposition 4. For , let . Then, the noncommuting graph of is given by

3. Construction of Vertex and Edge Partitions

First we define some useful parameters, which support in the investigation of some predefined (in Table 1) topological indices. For any vertex of G, these parameters are defined as follows:(i)The distance number of in G is (ii)The sum distance number of in G is (iii)The reciprocal distance number of in G is

According to these parameters, the distance-based topological indices, listed in Table 1, become

Let be a commuting graph of the dihedral group . In , there are vertices. The number of edges in is when n is odd and is when n is even. Based on the degree, distance number, sum distance number, and reciprocal distance number of each vertex of , the useful vertex partition is given in Table 2. Based on degrees and degrees sum of the end vertices of each edge of , the useful edge partition is given in Table 3.

Let be a non-commuting graph of the dihedral group . In ,(1)There are vertices and edges when n is odd(2)There are vertices and edges when n is even

Based on the degree, eccentricity, distance number, sum distance number, and reciprocal distance number of each vertex of , the useful vertex partition is given in Table 4. Based on degrees and degrees sum of the end vertices of each edge of , the useful edge partition is given in Table 5.

4. Topological Properties of Commuting Graph

In this section, we compute the Wiener, reciprocal complementary Wiener, MTI, Harary general Randić, , , , , and harmonic indices of . Throughout this section, in each of the two-row equation arrays, the first row corresponds to odd values of n, while the second corresponds to even values of n.

Theorem 1. For , let be a commuting graph on , then

Proof. Using the vertex partition, given in Table 2, in formula (6) of the Wiener index, we haveNow, the required Wiener index can be obtained after some simplifications.

Theorem 2. For , let be a commuting graph on , then

Proof. Since the diameter of is 2, so by using the vertex partition, given in Table 4, in formula (7) of the reciprocal complimentary Wiener index, we haveNow, the required index can be easily found by performing some simplifications.

Theorem 3. For , let be a commuting graph on , then

Proof. By applying formula (8) of the Schultz molecular topological index using the vertex partition, given in Table 2, we have when n is oddand when n is even

Theorem 4. For , let be a commuting graph on , then

Proof. Using the vertex partitions, given in Table 2, in formula (9) of the Harary index, we haveSome easy simplifications yield the required Harary index.

Theorem 5. For , let be a commuting graph of . Then, for odd values of n,and for even values of n,

Proof. Using the edge partition, given in Table 3, in the formula of general Randić index for , we haveAfter a minor simplification, we get our required result.

Theorem 6. For , let be a commuting graph of , then

Proof. Applying formulas of the geometric arithmetic index and its fifth version, using the edge partition given in Table 3, we haveThe required values of the geometric arithmetic index and its fifth version can be obtained after some simplifications.

Theorem 7. For , let be a commuting graph of , then

Proof. By using the edge partition, given in Table 3, in formulas of and indices, we haveAlso, for odd values of n, we haveand for even values of n, we haveThe required formulas for both the indices one can get by performing an easy simplification.

Theorem 8. For , let be a commuting graph of , then

Proof. By applying the formula of the harmonic index, using the edge partition given in Table 3, we haveSome simplifications yield the required values of the harmonic index.

5. Topological Properties of Noncommuting Graph

In this section, we compute the reciprocal complementary Wiener, Harary, general Randić, , , , , and harmonic indices of .

Theorem 9. For , let be a noncommuting graph of , then

Proof. Since the diameter of is 2, so by sing the vertex partition, given in Table 4, in formula (7) of the reciprocal complimentary Wiener index, we haveExact values for this index are due to some easy calculations.

Theorem 10. For , let be a noncommuting graph of , then

Proof. By sing the vertex partition, given in Table 4, in formula (9) of the Harary index, we haveBy performing some algebraic computations, one can obtain the required Harary index.

Theorem 11. For , let be a noncommuting graph of . Then, for odd values of n,and for even values of n,

Proof. Using the edge partition, given in Table 5, in the formula of general Randić index for , we haveBy performing some simplifications, we get the required results.

Theorem 12. Let be a noncommuting graph of , for , then

Proof. Applying formulas of the geometric arithmetic index and its fifth version, using the edge partition given in Table 5, we haveThe required values of the geometric arithmetic index and its fifth version can be obtained after some simplifications.

Theorem 13. For , let be a noncommuting graph of , then

Proof. By using the edge partition, given in Table 5, in formulas of and indices, we haveAlso, for odd values of n, we haveand for even values of n, we haveThe required formulas for both the indices one can get by performing an easy simplification.

Theorem 14. For , let be a noncommuting graph of , then

Proof. By applying the formula of the harmonic index, using the edge partition given in Table 5, we haveSome simplifications yield the required values of the harmonic index.

6. Concluding Remarks

An algebraic structure plays a vital role in chemistry to form chemical compound structures and in investigating various chemical properties of chemical compounds in these structures. Here, we considered a very well-known algebraic structure, called the group of symmetries of regular gons (the dihedral group), which has remarkable contribution in the theory of electron structures and molecular vibrations. We considered one algebraic property, namely, commutation property, on the dihedral group and associated two graphs (chemical structure) with the group of symmetries. We computed some distance-based and degree-based topological properties of these associated graphs by computing the exact formulae of the Wiener index, reciprocal complementary Wiener index, Schultz molecular topological index, Harary index, Randić index, geometric arithmetic indices, atomic bond connectivity indices, and harmonic index. All the indices are numeric quantities and, in fact, this work is a theoretical contribution in the theory of topological indices with the unique algebraic structure, and it can be very helpful to predict the bioactivity of chemical compounds using physicochemical properties in QSAR/QSPR studied.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by Top-Notch Talents Cultivation Project of Anhui Higher Education (Grant no. gxyq2017081) and Natural Science Fund of Education Department of Anhui Province (Grant no. KJ2017A4691).

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Copyright © 2020 Chang-Cheng Wei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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