Schultz and Modified Schultz Polynomials of Cog-Complete Bipartite Graphs

Let G be a simple connected graph, the vertexset and edgeset of G are denoted by V(G) and E(G), respectively. The molecular graph G, the vertices represent atoms and the edges represent bonds. In graph theory, we have many invariant polynomials and many invariant indices of a connected graph G. Topological indices based on the distance between the vertices of a connected graph are widely used in theoretical chemistry to establish relation between the structure and the properties of molecules. The coefficients of polynomials are also important in the knowledge some properties in application chemistry. The Schultz and modified Schultz polynomials, Schultz and modified Schultz indices and average distance of Schultz and modified Schultz of Cog-complete bipartite graphs are obtained in this paper.


Introduction
Suppose that = ( ( , ( is a simple undirected connected graph of order = ( and size = ( . A graph G is called an n-partite graph, ≥ 2, if it possible to partition vertex-set ( in to non empty subsets , , … , (called partite sets) such that every element of ( joins a vertex of V to a vertex of V , for all r ≠ t for n = 2, such graphs are called bipartite graphs. A complete bipartite graph G is a 2-partite graph with petite sets V and V having the added property that if ∈ and ∈ , then ∈ ( with two partite sets V and V denoted by K , (or K(m, n , where |V | = m and |V | = n . The distance between any two vertices u and v of G is the length of a shortest (u,v)-path in a connected graph G, denoted by ( , . In particular, if = , then ( , = 0 .
Topological indices in biology and chemistry was used for the first time in 1947 when chemist Harold Wiener [1] introduced Wiener index to demonstrate correlations between physicochemical properties of organic compounds of molecular graphs. The Wiener index is represent the total distance of a connected graph G, i.e. "( = ∑ ( , $,%∈&(' , in which the summation is taken over all unordered pairs { , } of distinct vertices of G. The diameter of G is the greatest distance in G and will be denoted by *. The number of pairs of vertices of G that are distance k is denoted by ( , + . Distance is an important concept in graph theory and it has applications to computer science, chemistry, and a variety of other fields [2][3][4][5][6].
For the definitions of concepts and notations used in this papers, vertices with distance k such that |, -( | = ( , . These based structure descriptors and their polynomials were extensively studied and computed before [7][8][9]. The molecular topological index (Schultz index) was introduced by Harry P. Schultz in 1989 [10] and the modified Schultz index was defined by S. Klavz ?ar and I. Gutman in 1997 [11]. In 2005, Gutman find many relations between Hosoya, Schultz and modified Schultz polynomials of a tree graph and some properties [12]. Bo Zhou [13], find some lower and upper bounds for the Schultz index of a graph G.
The Schultz and modified Schultz polynomial of some special graphs are summarized in the following theorem (See [14]). In addition, we have found the Schultz and modified Schultz polynomials of some Cog-special graphs which it under publication.
In 2013, Hassani et al. computed the Schultz polynomials of isomeres of C EE Fullerene by GAP program [15]. The Schultz indices have been shown to be a useful molecular descriptors in the design of molecules with desired properties, reader can be see the papers [16][17][18].