On general [formula omitted]-type index of connected graphs

Abstract

For a simple connected graph G=(V,E), define ABCα(G;β)=∑uv∈EdG(u)+dG(v)−βdG(u)dG(v)α,for any α∈R∖{0} and β∈R. This topological index generalizes the famous ABC index, augmented Zagreb index, and inverse sum indeg index. Let Γ(π) be the class of connected graphs with given degree sequence π. In this paper, under different conditions of α and β, we show that (i) For any connected graph G with uv∉E(G), ABCα(G+uv;β)>ABCα(G;β) holds; (ii) For any given degree sequence π with minimum degree 1, there exists a special extremalBFS-graph (see Definition 1.2) for ABCα(G;β) in the class of Γ(π); (iii) For any given c-cyclic degree sequence π with minimum degree 1 and c∈{0,1,2}, there exists a precise maximum extremalBFS-graph (see Definition 1.3) for ABCα(G;β) in the class of Γ(π). These extend the main results of (Chen et al., 2021; Lin et al., 2013; Lin et al., 2018; Chen and Hao, 2018).