Hosoya properties of the commuting graph associated with the group of symmetries


 A vast amount of information about distance based graph invariants is contained in the Hosoya polynomial. Such an information is helpful to determine well-known distance based molecular descriptors. The Hosoya index or Z-index of a graph G is the total number of its matching. The Hosoya index is a prominent example of topological indices, which are of great interest in combinatorial chemistry, and later on it applies to address several chemical properties in molecular structures. In this article, we investigate Hosoya properties (Hosoya polynomial, reciprocal Hosoya polynomial and Hosoya index) of the commuting graph associated with an algebraic structure developed by the symmetries of regular molecular gones (constructed by atoms with regular atomic-bonding).


Introduction
A numeral quantity which captures the symmetry of a molecular structure is known as a topological index. In fact, a topological index is a numerical characterization of a chemical graph and it provides a mathematical function of the structure in quantitative structure-activity relationship (QSAR)/quantitative structure-property relationship (QSPR) studies. It correlates certain physico chemical properties such as boiling point, stability and strain energy of chemical compounds of a molecular structure (graph). Several properties of chemical compounds in a molecular structure can be determined with the assistance of mathematical languages rendered by various types of topological indices. Harold Wiener, introduced the concept of first (distance based) topological index while working on the boiling point of Paraffin and named this index as path number (Wiener, 1947). Next off, it was renamed as the Wiener index, and that was the time theory of topological indices began.
We consider simple and connected graph (chemical structure) G with vertex set V(G) and edge set E(G). We denote the two adjacent vertices u and v in G as ∼ u v and non-adjacent vertices as  .
u v Starting from a vertex u and ending at a vertex v in G, a shortest alternating sequences of vertices and edges without repetition of any vertex is known as a − u v geodesic. The number of edges in a − u v geodesic is denoted by ( , ), d u v and is called the distance between u and v in G. The sum of two graphs G 1 and G 2 , denoted by G 1 + G 2 , is a graph with vertex set ∪ 1 2 ( ) ( ) V G V G and an edge set The maximum distance of a vertex v among all of its distances with all the vertices of G is called the eccentricity of , v denoted by ( ). ecc v The number

Hosoya properties
Many chemist used the concept of counting polynomial, introduced by Polya (1936), in order to obtain the molecular orbitals of unsaturated hydrocarbons. In this regard, the spectra of the characteristic polynomial of graphs were studied extensively. In 1988, Hosoya used this concept to find the polynomials of many chemical structures (Hosoya, 1986b), known as Hosoya polynomials and seeks a lot of attention afterwards. In 1996, the Hosoya polynomial was renamed as Wiener polynomial by Sagan et al. (1996), but now a days, majority of researchers used the term Hosoya polynomial. The Hosoya polynomial provides a pile of information about distance based graph invariants. The relationship between the Hosoya polynomial and the hyper Wiener index was observed by Cash (2002). Several other applications of extended Wiener indices were studied by Estrada et al. (1998).
Consider a connected graph G of n vertices. Hosoya defined the polynomial of G with variable x in the following way: is called the reciprocal status (transmission) of a vertex . v The Hosoya index or Z-index of a graph G is the total number of its matchings (independent edge subsets, the empty set is also consider one matching in G). The Hosoya index is the prominent example of topological indices which is of great interest in combinatorial chemistry. The Z-index was introduced by Hosoya (1971Hosoya ( , 1986a and it turned out to be applicable to molecular chemistry, such as boiling point, entropy or heat of vaporization are well studied. Several researcher studied extremal problems with respect to the Hosoya index for various graph structures. With respect to Hosoya and Merrifield-Simmons indices, extremal properties of various graph, trees and unicyclic graph have been studies in the articles of Deng and Chen (2008), Wagner (2007), and Yu and Tian (2006). The minimal Hosoya index for cyclic systems was computed by Hou (2002). In 2008, Deng investigated the largest Hosoya index for every simple connected garph having n vertices and + 1 n edges (Deng, 2008).

Group of symmetries and commuting graph
Group of symmetries finds its notable use in the theory of molecular vibrations and electron structures. Due to their noteworthy employment in chemical structures, in the context of topological indices, we consider the group of symmetries of a regular molecular polygon (also called a regular molecular n-gon constructed on ≥ 2 n atoms with their regular atomic-bonding). A regular molecular n-gon is a molecular structure whose corners are atoms and sides are atom-bonds of same length, and each internal angle between atom-bonds is of the measurement 2k n π π − radian. The group of symmetries of a regular molecular n-gon consists of 2n elements, which are n rotations about its center through an angle of 2k n π radian, where = −  0,1, , 1 k n , either all clockwise or all anti-clockwise) and n reflections (for even n, the reflections through a line joining the mid-points of the opposite atom-bonds or through a line joining two opposite atoms; and for odd n, the reflections through those lines which join an atom with the mid-point of the opposite atom-bond). Symbolically, the group of symmetries is denoted by D n and is called the dihedral group (a group theoretic name) of order 2n (Majeed, 2013). If we denote a rotation by 'a' and a reflection by 'b', then 2n elements of D n are The center of D n is: The commuting graph of a non-abelian group Γ is denoted by ( , ) . Ω ⊆ Γ For two unlike elements , , x y x y ∈Ω ∼ in G Γ if and only if = xy yx in Γ . The notion of commuting graphs on non-central elements of a group has been studied by many researchers -see for instance Ali et al. (2016) and Bunday (2006), and the references therein. The commuting graph on n D is defined by Ali et al. (2016) in the following result. , when is odd, , when is even. 2 Here K 1 is the trivial graph, K p is a complete graph on p vertices, N t is a null graph on t vertices, 2 2 n K is the union of 2 n copies of 2 . K

Hosoya polynomials
In this section, we find Hosoya polynomial and reciprocal status Hosoya polynomial of G Γ .

Hosoya polynomial
Our first two results of this section provide the coefficients for the Hosoya polynomial of the commuting graph on . n D

Proposition 2
Consider the commuting graph G Γ associated with the group n D Γ = for odd values of ≥ 3 n . Then: Consider the set V p of all the pairs of vertices (same and distinct) of G Γ , then: ) . V N Ω = Ω Therefore:

Proposition 3
Consider the commuting graph G Γ associated with the group n D Γ = for even values of ≥ 4 n . Then: If p V denotes the set of all the pairs of vertices (same and distinct) of G Γ , then | | (2 1), The Hosoya polynomials of the commuting graph on , n D for odd and even values of ≥ 3, n is obtained in the following result.

Reciprocal status Hosoya polynomial
Firstly, we find the reciprocal status of each vertex of the commuting graph on D n , for odd and even values of ≥ 3, n in the following two results, respectively.

Proposition 5
If l is a vertex in the commuting graph on D n for odd values of ≥ 3 n , then: The next two results provide the reciprocal status Hosoya polynomial of the commuting graph on D n .

Theorem 7
For odd ≥ 3 n , if G Γ is the commuting graph on . n D Γ = Then: Proof. Proposition 5 implies that there are three types ( ∼ a b, ∼ a c, ∼ b b) of edges in G Γ according to the reciprocal statuses of their end vertices, where 2 1, a n = − 3 1 2 Table 1 shows the edge partition accordingly.
By using the edge partition, given in the Table 1, in the formula of the reciprocal status Hosoya polynomial, we have: ) of edges in G Γ accordingly, the edge partition is given in the Table 2 By using the edge partition of G Γ , given in the

Hosoya index
In this section, we investigate the Hosoya index of the commuting graph of the dihedral group. The largest possible value of the Hosoya index, on a graph with n vertices, is given by the complete graph K n (Tichy and Wagner, 2005). Generally, the Hosoya index of a complete graph K n , ≥ 1 n is:    ∪ Ω = Ω , so the number of matchings in this type can be obtained by counting the matchings in K n for all ≥ 3, n which are given in Table 4, where m k denotes the number of matchings of order k for 1 [ ] 2 n k ≤ ≤ .
(M 2 ) Each matching of this type can be obtained by adding one edge of Type-3 into each matching of the edges of Type-1. As each edge of Type-1 is an edge of a complete graph −1 n K induced by the vertices in 3 Ω , so each matching of the edges of Type-1 is actually the matching in a complete graph −1 n K . The numbers of such matchings are listed in Table 5, where m k denotes the number of matchings of order k for 1 [ ] 2 n k ≤ ≤ .
Now, since there are n edges of Type-3, so the required matchings can be obtained as follows: Matchings of order 1: These are n such matchings corresponding to n edges of Type-3.
Matchings of order 2: Each of these matchings can be obtained by adding one edge of Type-3 into each matching of order 1 in −1 n K . There are n edges of Type-3 and 1 ( ) 2 n − matchings of order 1 in −1 n K , by Table 5. Therefore, by the rule of product, the number of matchings of order 2 is:  Table 5.

Matchings
Therefore, by the rule of product, the number of matchings of order 3 is:

Matchings of order 4:
Each of these matchings can be obtained by adding one edge of Type-3 into each matching of order 3 in −1 n K . There are n edges of Type-3 and matchings of order 3 in −1 n K , by Table 5. Therefore, by the rule of product, the number of matchings of order 4 is:  Table 5. Therefore, by the rule of product, the number of matchings of order k is: Now, by the rule of sum, the total number of matchings (matchings (M 1 ) + matchings (M 2 )) in G Γ of each order can be counted as follows: The number of matching of order 1 is: The number of matching of order 2 is: The number of matching of order 3 is: Generally, the number of matching of order k is: n then 2 D Γ = is an abelian group and so the commuting graph G Γ is the complete graph 4 . K Hence, by Table 3, the Hosoya index of G Γ is + + =