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

The hyper-Zagreb index of a simple connected graph G is defined by χ 2 ( G ) = ∑ u v ∈ 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 M 1 ( G ) , second Zagreb index M 2 ( G ) , harmonic index H ( G ) , and inverse edge degree I E D ( G ) . In addition, we give the identities on Hyper-Zagreb index and its coindex for the simple connected graphs.