The hyper-Zagreb index of cacti with perfect matchings

Abstract

Let G be a simple connected graph. The hyper-Zagreb index is defined as HM(G)=∑uv∈E(G)(dG(u)+dG(v))2. A connected graph G is a cacti if all blocks of G are either edges or cycles. Let ζ(2n,r) be the set of cacti of order 2n with a perfect matching and r cycles. In this paper, we determine sharp upper bounds of the hyper-Zagreb index of cacti among ζ(2n,r) and characterize the corresponding extremal cacti.