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Maximum Zagreb index, minimum hyper-Wiener index and graph connectivity. (English) Zbl 1171.05350

Summary: We show that among all \(n\)-vertex graphs with edge or vertex connectivity \(k\), the graph \(G=K_k\vee (K_{1}+K_{n-k-1})\), the join of \(K_k\), the complete graph on \(k\) vertices, with the disjoint union of \(K_{1}\) and \(K_{n-k-1}\), is the unique graph with maximum sum of squares of vertex degrees. This graph is also the unique \(n\)-vertex graph with edge or vertex connectivity \(k\) whose hyper-Wiener index is minimum.

MSC:

05C40 Connectivity
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