Certain Topological Indices of Non-Commuting Graphs for Finite Non-Abelian Groups

A topological index is a number derived from a molecular structure (i.e., a graph) that represents the fundamental structural characteristics of a suggested molecule. Various topological indices, including the atom-bond connectivity index, the geometric–arithmetic index, and the Randić index, can be utilized to determine various characteristics, such as physicochemical activity, chemical activity, and thermodynamic properties. Meanwhile, the non-commuting graph ΓG of a finite group G is a graph where non-central elements of G are its vertex set, while two different elements are edge connected when they do not commute in G. In this article, we investigate several topological properties of non-commuting graphs of finite groups, such as the Harary index, the harmonic index, the Randić index, reciprocal Wiener index, atomic-bond connectivity index, and the geometric–arithmetic index. In addition, we analyze the Hosoya characteristics, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of the non-commuting graphs over finite subgroups of SL(2,C). We then calculate the Hosoya index for non-commuting graphs of binary dihedral groups.


Introduction
In a broad sense, molecular descriptors are a method for describing and quantifying a chemical composition using mathematics and cheminformatics techniques. It is necessary to understand that no molecular descriptor applies to all applications. Different descriptors can be used to study and describe the same molecule, depending on the question to be answered and the goals to be reached. There are several types of molecular descriptors, some of which use chemical graph theory [1]. These include chemical indices, topological indices, autocorrelation descriptors, geometrical descriptors, and certain molecular fingerprints. Most of them are useful for Computer-Assisted Structure Elucidation (CASE): to evaluate the topology and geometry between the data source and desired molecules; to easily determine identical features between a large number of chemical graphs, and to enable rapid scanning of chemical libraries based on essential molecular characteristics. Topological indices are two-dimensional molecular descriptors depending on the graph representation of the topology of the molecular structure. The molecular graph is the first topological index, representing a molecule in two dimensions. The first topological index is the molecular graph, which is a 2D graph that shows how a molecule appears. The molecular graph is a sparse, undirected, and weighted multigraph. When a chemical structure is shown as a graph, well-known tools from graph theory can be used to find three dimensions are calculated. In addition, they provided examples of dimensionless commuting involution graphs. In [28], the authors investigated the Hosoya characteristics of non-commuting graphs of dihedral groups. In [5], the authors analyzed the Hosoya properties of power graphs of various finite groups. Several types of topological indices have been applied to commuting graphs related to finite groups, for instance, in [29,30], while the authors of [31] studied several topological indices of the non-commuting graphs over dihedral and generalized quaternion groups, respectively. Motivated by their work (as mentioned above), we devote ourselves to the non-commuting graphs of finite subgroups SL(2, C). It is very complicated to calculate the topological indices of Γ G for any finite group G. So, in this article, we focus our attention to examine several topological indices (as stated in Table 1), the (reciprocal status) Hosoya polynomial and the Hosoya index of a finite groups.
There are still significant gaps in the existing work about the identification of certain topological properties, the (reciprocal status) Hosoya polynomials as well as the Hosoya index of non-commuting graphs of finite subgroups of SL(2, C). The apparent explanation is that neither the construction of non-commuting graphs over finite groups nor the derivation of handy formulas of graph characteristics for comprehensive classes of groups. We make an attempt in this article to examine one of these problems This article is structured as follows: Section 2 covers some findings and essential definitions that are useful to this article. Section 3 explores the construction of edge and vertex partitions. Various topological properties of non-commuting graphs over binary dihedral groups are demonstrated in Section 4. Section 5 discusses the Hosoya properties, that is, the Hosoya and its reciprocal status, and the Hosoya index of the non-commuting graph for finite subgroups of SL(2, C). Section 6 contains the conclusion and future work of the article.

Preliminaries
This section summarizes numerous basic graph-theoretic features and notable results that will be discussed in more detail later in this paper.
Assume that Γ is an undirected simple graph. The edge and vertex sets of Γ are denoted by E(Γ) and V(Γ), respectively. The order of Γ is the total number of vertices represented by |Γ|. The distance between vertices u 1 and u 2 in Γ, denoted by dis(u 1 , u 2 ), is defined as the shortest path in both nodes u 1 and u 2 . Two vertices v 1 and v 2 are connected if they share an edge, and it is represented by v 1 ∼ v 2 , otherwise v 1 v 2 . N(x) represents the neighborhood of x, which consists of all vertices in Γ adjacent to x. The degree (valency) denoted by d u 1 of u 1 is the set of vertices in Γ, that are edge connected to u 1 , and the degree sum of a vertex u is S u = ∑ v∈N(u) d v . A u 1 − u 2 path having dis(u 1 , u 2 ) length is known as a u 1 − u 2 geodesic. The greatest distance between u 1 and any other vertex in Γ is referred to as the eccentricity, and it is represented by ec(u 1 ). Amongst every vertex in Γ, the diameter denoted by diam(Γ) has the highest eccentricity. Additionally, amongst every vertex of Γ, the radius rad(Γ) has the smallest eccentricity. Furthermore, a vertex u 1 is said to be a central vertex of Γ, if ec(Γ) = rad(Γ) and a vertex u 1 is called peripheral vertex, if ec(Γ) = diam(Γ). A subgraph induced by the central vertices and peripheral vertices of Γ are called centre and periphery, respectively. A graph Γ is known as self-centered if rad(Γ) = diam(Γ).
Suppose Γ 1 and Γ 2 are two connected graphs, then Γ 1 ∨ Γ 2 is the join of Γ 1 and Γ 2 whose edge and vertex sets are A complete graph is one in which each individual vertex in the graph has an edge, and it is denoted by K n . A graph that has its vertices partitioned into k different independent sets is said to be k-partite, and a complete k-partite graph contains an edge between any two vertices from different independent sets. Additional undefined expressions and symbols were obtained from [39].

Definition 1.
Assume that G is a group. Then the centre of G is described as follows: As ec(u) ≤ 2 for each u ∈ Γ G , so we have the following proposition.
where G is a non-abelian group, and if for each u ∈ Γ G , we have ec(u) = 2. However, it is equal to the sum of the periphery and the center of Γ G .
The number of conjugacy classes in a group G is represented by the symbol κ(G), while Z n is used to denote the cyclic group of order n. The set of 2 × 2 matrices whose determinant is one forms the special linear group SL(2, C) of degree 2 over the complex field C. Moreover, the presentation of the binary dihedral group BD 4n of order 4n is defined as: We now divide BD 4n as follows: Therefore, there are n + 3 conjugacy classes of BD 4n . Furthermore, represents the binary tetrahedral group of order 24, the binary octahedral group of order 48, and the binary icosahedral group of order 120, respectively. All the mentioned above are finite non-abelian subgroups of SL(2, C). Several characteristics of the mentioned groups will be investigated, but the noncommuting graph of BD 4n will be our prime motive. Hence, using GAP [40] calculations, we obtain Propositions 1 and 2, so we deduce the subsequent result, that is, the classification of the non-commuting graphs of finite subgroups of SL(2, C). Proposition 3. The non-commuting graphs of finite subgroups of SL(2, C) have the following structure: According to the above classification, we obtain the following points of the noncommuting graph Γ BD 4n of BD 4n : 1. For It can be observed in Γ BD 4n that ec(w 2 ) = 2 for every w ∈ X 2 ∪ X 3 . As a result, Γ BD 4n is a self-centered graph, that is equivalent to K 2, 2, . . . , 2 where 0 ≤ i ≤ n − 1, and one partite set X 3 .
The following relevant properties for the non-commuting graph Γ G was suggested in [13,30].

Edge and Vertex Partitions
To begin, we develop a number of interesting components that help in the evaluation of certain topological indices. The following parameters are defined for each u of Γ:

Topological Properties
Several topological characteristics of non-commuting graphs associated with binary dihedral groups are discussed in this section. Theorem 1. Suppose Γ BD 4n is a non-commuting graph of BD 4n . Then: Proof. We have determined the Harary index by substituting the vertex partition, as mentioned in Table 2 and in Equation (3).
One may derive the required Harary index using a series of algebraic calculations.
Theorem 2. Assume that Γ BD 4n is a non-commuting graph of BD 4n . Then: Proof. By applying the edge partition presented in Table 3 and the harmonic index in Table 1, we obtain: Certain computations result in the appropriate formula for the harmomic index. Table 3. Edge partition of Γ BD 4n for any u ∼ w ∈ E(Γ BD 4n ).
Note that (d v , d w ) represents the kind of v ∼ w edge defined by the degrees of the end vertices, while (S v , S w ) represents the kind of v ∼ w edge defined by the degrees sum of the end vertices.
Theorem 3. Suppose Γ BD 4n is the non-commuting graph of BD 4n Then: , Proof. Compute the edge partition, as shown in Table 3, using the generic Randić index R α formula for α = 1, −1, 1 2 , − 1 2 , we have: We obtain the desired results by applying certain simplifications.

Theorem 4.
Suppose Γ BD 4n is a non-commuting graph of BD 4n . Then Proof. As Γ BD 4n has a diameter of 2, so by computing the RCW index, we may use Equation (1) and the vertex partition as shown in Table 2.
We get the desire result by applying certain simplifications.

Theorem 5.
Assume that Γ BD 4n is a non-commuting graph of BD 4n . Then: Proof. By incorporating the edge partition presented in Table 3 into the ABC as well as ABC 4 indices calculations, we obtain: , and One may have the appropriate formulae for both indices by performing a simple simplification.
Theorem 6. Assume that Γ BD 4n is the non-commuting graph of BD 4n . Then: Proof. We have obtained the geometric arithmetic GA index and its 5th version by utilizing the formulae and the edge partition in Table 3.
After several computations, the needed values of the GA index and its 5th version may be achieved.

Hosoya Properties
The next section defines the Hosoya properties that are being considered and calculate them for finite subgroups of SL(2, C). We begin by computing the Hosoya polynomial, then determine its reciprocal status, and at last, we explore the Hosoya index.

Hosoya Polynomial
The first two results in this subsection give the coefficients required to build the Hosoya polynomial for the non-commuting graph on BD 4n .
The following result determines the Hosoya polynomials of Γ BD 4n .

Theorem 7.
For any n ≥ 2, the Hosoya polynomial of Γ BD 4n is given as: Proof. Using the values of dis Γ BD 4n , u from Propositions 4, we obtain the following formula presented in Table 1 for the Hosoya polynomial: Theorem 8. Assume that Γ G is the non-commuting graph of G. Then: If G = BO 48 , then H(Γ G , x) = 75x 2 + 960x + 46.
Proof. Using Proposition 3, GAP [40], and applying the same calculations as in Theorem 7, we can obtain the desired result.

Reciprocal Status Hosoya Polynomial
To begin, we determine the reciprocal status of every vertex of the non-commuting graphs of finite subgroups of SL(2, C). Then discuss its reciprocal status Hosoya polynomial.
Theorem 9. For any n ≥ 4, the reciprocal status Hosoya polynomial of Γ BD 4n is given by: Proof. According to Proposition 5, Γ BD 4n has two kinds of edges (α ∼ β and α ∼ β) based on the reciprocal status of end vertices, whenever α = 8n−7 2 while β = 3(2n−1) 2 . The reciprocal Hosoya polynomial's formula is provided in Table 1, and we can use the edge partition from Table 4 to obtain: This conclusively establishes the proof. Table 4. Edge partition of Γ BD 4n for any x ∼ y ∈ E(Γ BD 4n ).

Kind of Edge Partition of the Edge Set Counting Edges
Theorem 10. Assume that Γ G is the non-commuting graph of G. Then: 39 .
Proof. Using Proposition 5, GAP [40], and applying the same calculations as in Theorem 9, we can obtain the desired result.

Hosoya Index
The Hosoya index of the non-commuting graph on BD 4n is investigated in this subsection. To begin, take a note of the total of non-empty matchings presented in Table 5 for K m , whereas δ τ represents the total possible matchings of order τ, where 1 ≤ τ ≤ n. Table 5. Non-empty K m matchings.
For any n ≥ 2, the subsequent result calculates the Hosoya index of Γ BD 4n on BD 4n .
Theorem 11. The Hosoya index of Γ BD 4n is given as: where δ 1 1 = 4n(n − 1) and for 2 ≤ τ ≤ 2n − 2, Proof. As a result of Proposition 3, it is clear that, As a result, Γ BD 4n has the succeeding two kinds of edges.
As a consequence, there are 3 distinct forms of matchings among the Γ BD 4n edges. M 3 : Type-1 and Type-2 edge matchings M 1 : If δ 1 τ determines the number of order τ matchings, then the total possible order 1 matchings equals the number of Type-1 edges, which is 4n(n − 1), i.e., δ 1 1 = 4n(n − 1). Furthermore, we have In general, for any 2 ≤ τ ≤ 2(n − 1), Note that Thus, for 2 ≤ τ ≤ 2(n − 1), M 2 : If δ 2 τ signifies the total possible matchings of cardinality τ, then the total possible matchings of cardinality 1 equals the total Type-2 edges. It is worth noting that the Type-2 edges correspond to the edges of Γ BD 4n 's subgraph K 2, 2, . . . , 2 n−times , which is isomorphic to K 2n − ne, where K 2n − τe represents a graph formed by removing τ edges from K 2n . Therefore, the size of K 2, 2, . . . , 2 n−times is equal to the size of K 2n − ne, In this paper, we tried to investigate several topological properties of non-commuting graphs over finite subgroups of SL(2, C); specifically, the binary dihedral groups. However, the problem of determining the topological properties of (non-) commuting graphs, power graphs or Cayley graphs of any finite abelian or non-abelian group is still open and unresolved. An algebraic structure is essential for the development of chemical systems as well as the study of many chemical properties of molecules contained within these structures. Every index has a numerical value, and this work extends to topological indices with unique chemical structures, which may be beneficial for identifying bioactive compounds based on the physicochemical characteristics investigated in QSPR.

Data Availability Statement:
The data used to support the findings of this study are available within the article.