Abstract :
We investigate the index If (G) =(G) f(dG(v), dG(w)) of a graph G, where f is a symmetric function of two variables satisfying certain condi-tions, E(G) is the edge set of G, and dG(v) and dG(w) are the degrees of vertices v and w in G, respectively. Those conditions are satisfied by functions that can be used to define the general sum-connectivity index χa, general Randi¢ index Ra, general reduced second Zagreb index GRMa for some a ∈ R, general Sombor index SOa,b, general augmented Zagreb index AZIa,b and by one other generalization Ma,b for some a, b ∈ R. The general augmented Zagreb index is a new index defined in this paper. We obtain a sharp upper bound on If for graphs with given order and connectivity, and a sharp lower bound on If for 2-connected graphs with given order. Our upper bound holds for Ma,b and SOa,b where a, b ≥ 1; χa and Ra where a ≥ 1; and GRMa where a −1. Our lower bound holds for Ma,b where a ≥ 0 and b ≥ −a; SOa,b where a, b ≥ 0 or a, b ≤ 0; AZIa,b where a ≥ −2 and b ≥ 0; χa and Ra where a ≥ 0; and GRMa where a −2.
