On some topological indices of thorn graphs

Abstract

In this paper, the relation between the reciprocal Randic index, Reduced reciprocal Randic index and Atom-bond connectivity index of a simple connected graph and its thorn graph is stablished and the atom-bond connectivity \((A B C)\) index of a graph \(G\) is defined as \(A B C(G)=\sum_{w v \in E(G)} \sqrt{\frac{d_u+d_v-2}{d_u d_v}}\), where \(E(G)\) is the edge set and \(d_u\) is the degree of vertex \(u\) of \(G\) [13]. Reciprocal Randic \((R R)\) index of a graph \(G\) is defined as \(R R(G)=\sum_{w v \in E(G)} \sqrt{d_u d_v}\), where \(E(G)\) is the edge set and \(d_u\) is the degree of vertex \(u\) of \(G\). Reduced Reciprocal Randic \((R R R)\) index of a graph \(G\) is defined as \(\operatorname{RRR}(G)=\sum_{w v \in E(G)} \sqrt{\left(d_u-1\right)\left(d_v-1\right)}\), where \(E(G)\) is the edge set and \(d_u\) is the degree of vertex \(u\) of \(G\). Results are applied to compute the reciprocal Randic index, Reduced reciprocal Randic index and Atom-bond connectivity index of thorn rings, thorn paths, thorn rods, thorn star, thorn star \(S_n\left(p_1, p_2, \cdots, p_{n-1}, p_n\right)\).