Sharp Bounds of the Hyper-Zagreb Index on Acyclic, Unicylic, and Bicyclic Graphs

Abstract EN

The hyper-Zagreb index is an important branch in the Zagreb indices family, which is defined as ∑ u v ∈ E ( G ) ( d ( u ) + d ( v ) ) 2 , where d ( v ) is the degree of the vertex v in a graph G = ( V ( G ) , E ( G ) ) .

In this paper, the monotonicity of the hyper-Zagreb index under some graph transformations was studied.

Using these nice mathematical properties, the extremal graphs among n -vertex trees (acyclic), unicyclic, and bicyclic graphs are determined for hyper-Zagreb index.

Furthermore, the sharp upper and lower bounds on the hyper-Zagreb index of these graphs are provided.