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

Author

Listed:
  • 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
    University of Ljubljana
    FAMNIT, University of Primorska)

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).

Suggested Citation

  • Stéphane Bessy & François Dross & Katarína Hriňáková & Martin Knor & Riste Škrekovski, 2020. "The structure of graphs with given number of blocks and the maximum Wiener index," Journal of Combinatorial Optimization, Springer, vol. 39(1), pages 170-184, January.
    • Handle: RePEc:spr:jcomop:v:39:y:2020:i:1:d:10.1007_s10878-019-00462-6
    DOI: 10.1007/s10878-019-00462-6
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    References listed on IDEAS

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    1. Knor, Martin & Škrekovski, Riste, 2019. "On the minimum distance in a k-vertex set in a graph," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 99-104.
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    Cited by:

    1. Bessy, Stéphane & Dross, François & Hriňáková, Katarína & Knor, Martin & Škrekovski, Riste, 2020. "Maximal Wiener index for graphs with prescribed number of blocks," Applied Mathematics and Computation, Elsevier, vol. 380(C).

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