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Partition distance in graphs

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Abstract

If G is a graph and \(\mathcal{P}\) is a partition of V(G), then the partition distance of G is the sum of the distance between all pairs of vertices that lie in the same part of \(\mathcal{P}\). This concept generalizes several metric concepts and is dual to the concept of the colored distance due to Dankelmann, Goddard, and Slater. It is proved that the partition distance of a graph can be obtained from the Wiener index of weighted quotient graphs induced by the transitive closure of the Djoković–Winkler relation as well as by any partition coarser than it. It is demonstrated that earlier results follow from the obtained theorems. Applying the main results, upper bounds on the partition distance of trees with prescribed order and radius are proved and corresponding extremal trees characterized.

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Sandi Klavžar

  2. Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia

    Sandi Klavžar

  3. Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

    Sandi Klavžar

  4. Faculty of Science, Mahallat Institute of Higher Education, Mahallat, Islamic Republic of Iran

    M. J. Nadjafi-Arani

Authors
  1. Sandi Klavžar
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  2. M. J. Nadjafi-Arani
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Correspondence to M. J. Nadjafi-Arani.

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Klavžar, S., Nadjafi-Arani, M.J. Partition distance in graphs. J Math Chem 56, 69–80 (2018). https://doi.org/10.1007/s10910-017-0781-5

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