New bounds on the hyper-Zagreb index for the simple connected graphs.

Abstract

The hyper-Zagreb index of a simple connected graph G is defined by chi ^2(G) = sum_uv in E(G) left( d(u) + d(v) right)^2. In this paper, we establish, analyze and compare some new upper bounds on the Hyper-Zagreb index in terms of the number of vertices n, number of edges m, maximum vertex degree Delta, and minimum vertex degree delta, first Zagreb index M_1(G), second Zagreb index M_2(G), harmonic index H(G), and inverse edge degree IED(G). In addition, we give the identities on Hyper-Zagreb index and its coindex for the simple connected graphs.