On the maximum sigma index of k-cyclic graphs

Abstract

Let G be a graph with edge set E ( G ) . Denote by d w the degree of a vertex w of G . The sigma index of G is defined as ∑ u v ∈ E ( G ) ( d u − d v ) 2 . A connected graph of order n and size n + k − 1 is known as a connected k -cyclic graph. Abdo et al. (2018) characterized the graphs having the greatest sigma index over the family of all connected graphs of a fixed order. The primary goal of the present note is to determine the unique graph possessing the greatest sigma index from the class of all connected k -cyclic graphs of order n for every pair of fixed integers k and n satisfying 0 ≤ k ≤ n − 2 .