Lower bounds on the general first Zagreb index of graphs with low cyclomatic number

Abstract

The general first Zagreb index of a graph G, denoted by M1p(G), is defined as the sum of powers dGp(u) over all vertices u of V(G), where dG(u) denotes the degree of a vertex u in G. In this paper, we consider negative values of p and obtain sharp lower bounds on the general first Zagreb index of trees, unicyclic and bicyclic graphs in terms of their order and maximum vertex degrees. Also, the corresponding extremal graphs attaining the bounds are characterized. The results are then extended to other indices defined as sums over all vertices of contributions which are convex or concave functions of their degrees.