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 χ²(G) = ∑_{uv ∈ E(G)}(d(u) + d(v))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 Δ, and minimum vertex degree δ, first Zagreb index M₁(G), second Zagreb index M₂(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.