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Bounds and Relations Involving Adjusted Centrality of the Vertices of a Tree

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Abstract

Edge betweenness is a measure of the influence of edges in networks. The edge betweenness of an edge is defined to be the number of shortest paths between pairs of vertices that run along it. For a given graph \(G=(V(G),E(G))\), the adjusted betweenness centrality \(c_v^*\) of a vertex \(v\in V(G)\) is defined to be the sum of the edge betweenness of all the edges incident to \(v\). In this paper we study the mathematical properties of \(c_v^*\) for vertices in trees. We present lower and upper bounds for \(c_v^*\) in terms of the number of vertices, vertex degree, eccentricity, transmission, maximum degree, diameter, and radius.

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References

  1. Caporossi, G., Paiva, M., Vukicević, D., Segatto, M.: Centrality and betweenness: vertex and edge decomposition of the Wiener index. MATCH Commun. Math. Comput. Chem. 68, 293–302 (2012)

    MathSciNet  Google Scholar 

  2. Comellas, F., Gago, S.: Spectral bounds for the betweenness of a graph. Linear Algebra Appl. 423, 74–80 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Diestel, R.: Graph Theory Electronic Edition 2000. Springer, New York (1997)

    Google Scholar 

  4. Freeman, L.C.: A set of measures of centrality based on betweenness. Sociometry 40, 35–41 (1977)

    Article  Google Scholar 

  5. Freeman, L.C.: Centrality in social networks: conceptual clarification. Soc. Netw. 1, 215–239 (1978)

    Article  Google Scholar 

  6. Gago, S., Hurajovà, J., Madaras, T.: Notes on the betweenness centrality of a graph. Math. Slovaca 62, 1–12 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Girvan, M., Newman, M.E.J.: Community structure in social and biological networks. Proc. Natl. Acad. Sci. USA 99, 7821–7826 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Wiener, H.: Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)

    Article  Google Scholar 

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Acknowledgments

The authors are grateful to an anonymous referee for comments that improved the quality of the paper. This work was supported by GReGAS (European Union) and NSERC (Canada).

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Authors and Affiliations

  1. Department of Mathematics, University of Osijek, Osijek, Croatia

    Snježana Majstorović

  2. GERAD and HEC Montréal, Montreal, Canada

    Gilles Caporossi

Authors
  1. Snježana Majstorović
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  2. Gilles Caporossi
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Correspondence to Snježana Majstorović.

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Majstorović, S., Caporossi, G. Bounds and Relations Involving Adjusted Centrality of the Vertices of a Tree. Graphs and Combinatorics 31, 2319–2334 (2015). https://doi.org/10.1007/s00373-014-1498-x

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  • DOI: https://doi.org/10.1007/s00373-014-1498-x

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