On some properties of graph irregularity indices with a particular regard to the σ-index

Abstract

Some well-known graph irregularity indices of a connected graph G are investigated. Our study is focused mainly on the comparison of the Bell’s degree-variance (Var(G)) and the Collatz–Sinogowitz irregularity index (CS(G)) with the recently introduced σ(G) irregularity index. It is a degree-based topological invariant calculated as σ(G)=F(G)−2M2(G) where M2(G) is the second Zagreb index, F(G)=∑d3(v), and d(v) is the degree of the vertex v in G. By introducing the notion of the complete split-like graphs representing a broad subclass of bidegreed connected graphs, it is shown that for these graphs the equality σ(G)=n2Var(G) holds.