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Reverse Wiener indices of graphs of exactly two cycles. (English) Zbl 1256.05060

Summary: The reverse Wiener index of a connected graph \(G\) with \(n\) vertices is defined as \(\Lambda(G)= {1\over 2} n(n-1)d- W(G)\), where \(d\) and \(W(G)\) are respectively the diameter and Wiener index of \(G\).
We determine the \(n\)-vertex connected graph(s) of exactly two cycles of a vertex in common with the \(k\)th greatest reverse Wiener indices for all \(k\) up to three if \(n=7\), four if \(n=8\), \(\left\lfloor{\sqrt{n-7}\over 2}\right\rfloor+ 1\) if \(n\geq 9\), and the \(n\)-vertex connected graph(s) of exactly two vertex-disjoint cycles with the greatest reverse Wiener index. The \(n\)-vertex connected graphs with exactly two cycles with the greatest reverse Wiener index are determined for \(n\geq 7\).

MSC:

05C12 Distance in graphs
05C38 Paths and cycles
05C40 Connectivity
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