The structure of graphs with given number of blocks and the maximum Wiener index

Stéphane Bessy (), François Dross (), Katarína Hriňáková (), Martin Knor () and Riste Škrekovski ()
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Stéphane Bessy: Université de Montpellier
François Dross: Université de Montpellier
Katarína Hriňáková: Slovak University of Technology in Bratislava
Martin Knor: Slovak University of Technology in Bratislava
Riste Škrekovski: Faculty of Information Studies

Journal of Combinatorial Optimization, 2020, vol. 39, issue 1, No 12, 170-184

Abstract: 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 $$n-1$$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 $$p\ge 2$$p≥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 $$p=n-1$$p=n-1 for example).

Keywords: Graph theory; Wiener index; Distance (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10878-019-00462-6

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