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

The hyper-Zagreb index is an important branch in the Zagreb indices family, which is defined as ∑uv∈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