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

TL;DR

This paper introduces new status co-indices for graphs, explores their relationships with existing indices, and computes these indices for specific graph classes including intersection graphs, hypercubes, Kneser graphs, and achiral polyhex nanotorus.

Contribution

The paper defines novel status co-indices, establishes their relations with status connectivity indices, and computes these indices for various complex graph structures.

Findings

Relations between status connectivity indices and co-indices are established.

Explicit formulas for indices are derived for intersection graphs, hypercubes, Kneser graphs, and nanotorus.

New graph invariants are introduced and analyzed.

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.