Extremal augmented Zagreb index of trees with given numbers of vertices and leaves

Abstract

The augmented Zagreb index AZI(G) of a connected graph G is defined asAZI(G)=∑uv∈E(G)(d(u)d(v)d(u)+d(v)−2)3, where d(u) and d(v) are the degrees of the end-vertices of an edge uv, respectively. We determine the unique tree with given numbers of vertices and leaves that minimizes augmented Zagreb index, and characterize the unique graph with minimum augmented Zagreb index in the class of connected graphs with given numbers of vertices and pendent vertices. Furthermore, we also determine the maximum augmented Zagreb index with the extremal trees characterized in the class of all trees with given numbers of vertices and leaves.