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 Sigma(uv epsilon 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.