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Hyper- and reverse-Wiener indices of F-sums of graphs. (English) Zbl 1221.05120
Summary: The Wiener index \(W(G)=\sum _{\{u,v\} \subset V(G)}d(u,v)\), the hyper-Wiener index \(WW(G) = \frac{1}{2} \sum _{\{u,v\} \subset V(G)} [d(u, v) + d^2(u, v)]\) and the reverse-Wiener index \(\Lambda (G) = \frac{n(n-1)D}{2} - W(G)\), where \(d(u,v)\) is the distance of two vertices \(u,v\) in \(G, d^{2}(u,v)=d(u,v)^{2}, n=|V(G)|\) and \(D\) is the diameter of \(G\).
M. Eliasi and B. Taeri [“Four new sums of graphs and their Wiener indices,” Discrete Appl. Math. 157, No. 4, 794–803 (2009; Zbl 1172.05318)] introduced the F-sums of two connected graphs. In this paper, we determine the hyper- and reverse-Wiener indices of the F-sum graphs and, subject to some condition, we present some exact expressions of the reverse-Wiener indices of the F-sum graphs.

MSC:
05C12 Distance in graphs
05C76 Graph operations (line graphs, products, etc.)
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