Status connectivity indices and co-indices of graphs and its computation to intersection graph, hypercube, Kneser graph and achiral polyhex nanotorus

Abstract

The status of a vertex $u$ in a connected graph $G$, denoted by $\sigma_G(u)$, is defined as the sum of the distances between $u$ and all other vertices of a graph $G$. The first and second status connectivity indices of a graph $G$ are defined as $S_{1}(G) = \sum_{uv \in E(G)}[\sigma_G(u)+ \sigma_G(v)]$ and $S_{2}(G) = \sum_{uv \in E(G)}\sigma_G(u)\sigma_G(v)$ respectively, where $E(G)$ denotes the edge set of $G$. In this paper we have defined the first and second status co-indices of a graph $G$ as $\overline{S_{1}}(G) = \sum_{uv \notin E(G)}[\sigma_G(u)+ \sigma_G(v)]$ and $\overline{S_{2}}(G) = \sum_{uv \notin E(G)}\sigma_G(u)\sigma_G(v)$ respectively. Relations between status connectivity indices and status coindices are established. Also these indices are computed for intersection graph, hypercube, Kneser graph and achiral polyhex nanotorus.