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Erschienen in:
24.10.2019
The structure of graphs with given number of blocks and the maximum Wiener index
Erschienen in: Journal of Combinatorial Optimization | Ausgabe 1/2020
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Abstract
The Wiener index (the distance) of a connected graph is the sum of distances between all pairs of vertices. In this paper, we study the maximum possible value of this invariant among graphs on n vertices with fixed number of blocks p. It is known that among graphs on n vertices that have just one block, the n-cycle has the largest Wiener index. And the n-path, which has \(n-1\) blocks, has the maximum Wiener index in the class of graphs on n vertices. We show that among all graphs on n vertices which have \(p\ge 2\) blocks, the maximum Wiener index is attained by a graph composed of two cycles joined by a path (here we admit that one or both cycles can be replaced by a single edge, as in the case \(p=n-1\) for example).