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Steiner Wiener index of graph products. (English) Zbl 1463.05100

Summary: The Wiener index \(W(G)\) of a connected graph \(G\) is defined as \(W(G)=\sum_{u,v\in V(G)}d_G(u,v)\) where \(d_G(u,v)\) is the distance between the vertices \(u\) and \(v\) of \(G\). For \(S\subseteq V(G)\), the Steiner distance \(d(S)\) of the vertices of \(S\) is the minimum size of a connected subgraph of \(G\) whose vertex set is \(S\). The \(k\)-th Steiner Wiener index \(SW_k(G)\) of \(G\) is defined as \(SW_k(G)= \sum_{\overset{S\subseteq V(G)}{|S|=k}} d(S)\). We establish expressions for the \(k\)-th Steiner Wiener index on the join, corona, cluster, lexicographical product, and Cartesian product of graphs.

MSC:

05C09 Graphical indices (Wiener index, Zagreb index, Randić index, etc.)
05C12 Distance in graphs
05C75 Structural characterization of families of graphs
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