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.