Sharp bounds of the hyper-Zagreb index on acyclic, unicylic, and bicyclic graphs. (English) Zbl 1368.05030
Summary: The hyper-Zagreb index is an important branch in the Zagreb indices family, which is defined as \(\sum_{u v \in E(G)} (d(u) + d(v))^2\), 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.
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