K-Banhatti and K-hyper Banhatti Indices of Dominating David Derived Network

. Let G be connected graph with vertex V ( G ) and edge set E ( G ). The ﬁrst and second K-Banhatti indices of G are deﬁned as B 1 ( G ) = P ue [ d G ( u ) + d G ( e )] and B 2 ( G ) = P ue [ d G ( u ) d G ( e )], where ue means that the vertex u and edge e are incident in G . The ﬁrst and second K-hyper Banhatti indices of G are deﬁned as HB 1 ( G ) = P ue [ d G ( u ) + d G ( e )] 2 and HB 2 ( G ) = P ue [ d G ( u ) d G ( e )] 2 . In this paper, we compute the ﬁrst and second K-Banhatti and K-hyper Banhatti indices of Dominating David Derived networks. Mathematics


Introduction
Chemical graph theory is a branch of graph theory in which a chemical compound is represented by simple graph called molecular graph in which vertices are atoms of compound and edges are the atomic bounds. A graph is connected if there is atleast one connection between its vertices. Throughout this paper we take G a connected graph. If a graph does not contain any loop or multiple edges then it is called a network. Between two vertices u and v, the distance is the shortest path between them and is denoted by in graph G. For a vertex v of G the degree is number of vertices attached with it. The edge connecting the vertices u and v will be denoted by uv. Let d G (e) denote the degree of an edge e in G, which is defined by d G (e) = d G (u)+d G (v)−2 with e = uv. The degree and valence in chemistry are closely related with each other. We refer the book [1]for more details. Now a day another emerging field is Cheminformatics, which helps to predict biological activities with the relationship of Structure-property and quantitative structure-activity. Topological indices and Physico-chemical properties are used in prediction of bioactivity if underlined compounds are used in these studies [2,3]. A number that describe the topology of a graph is called topological index. In 1947, the first and most studied topological index was introduced by Weiner [4]. For more details about this index can be found in [5,6]. In 1975, Milan Randić introduced the Randić index [7]. Bollobas et al. [8] and Amic et al. [9] in 1998, working independently defined the generalized Randić index. This index was studied by both mathematicians and chemists [10]. For details about topological indices, we refer [11,12] The first and second K-Banhatti indices of G are defined as: where ue means that the vertex u and edge e are incident in G. The first and second K-hyper Banhatti indices of G are defined as and We refer [13] for details about these indices. The David derived and dominating David derived network of dimension n can be constructed as follows [14]: consider a n dimensional star of David network. Insert a new vertex on each edge and split it into two parts, we will get David derived network DD(n) of dimension n. The dominating David derived network of the first type of dimension n which can be obtained by connecting vertices of degree 2 of DDD(n) by an edge that are not in the boundary and is denoted by D 1 (n) [14].
The dominating David derived network of the second type of dimension n can be obtained by subdividing the new edges in D 1 (n) [14] and is denoted by Dominating David derived network of the second type D 2 (2) The dominating David derived network of the third type of dimension n denoted by D 3 (n) can be obtained from D 1 (n) by introducing a parallel path of length 2 between the vertices of degree two that are not in the boundary [14,15].
Proof. Let G = D 1 (n) be the dominating David derived network of 1 st type. From figure 1, the edge partition of dominating David derived network of 1 st type D 1 (n) based on degrees of end vertices of each edge is give in table 1. First Table 1. Edge partition of Dominating David derived network of first type Second K-Banhatti index of D 1 (n) is calculated as Proof. Let G = D 1 (n) be the dominating David derived network of 1 st type. Then first K-hyper Banhatti index of D 1 (n) is calculated as   Proof. Let G = D 2 (n) be the dominating David derived network of 2 nd type. Table 2 shows the edge partition of dominating David derived network of 2 nd type D 2 (n) based on degrees of end vertices of each edge 28n − 16 4 (3,4) 36n 6 − 56n + 24 5 (4,4) 36n 6 − 56n + 20 6 First K-Banhatti index of D 2 (n) is calculated as Second K-Banhatti index of D 2 (n) is calculated as Proof. Let G = D 2 (n) be the dominating David derived network of 2 nd type. First K-hyper Banhatti index of D 2 (n) is calculated as Second K-hyper Banhatti index of D 2 (n) is calculated as Proof. Let G = D 3 (n) be the dominating David derived network of 3 rd type. Table 3 shows the edge partition of dominating David derived network of 3 rd type D 3 (n) based on degrees of end vertices of each edge. First K-Banhatti index of D 3 (n) is calculated as  Second K-Banhatti index is calculated as = 94464n 2 − 130688n + 50688.

Conclusion
In the present report, we have computed First and second K-Banhatti and K-hyer Banhatti indices of Dominating David derived networks of first, second and third type.

Competing Interests
The author do not have any competing interests in the manuscript.