On the Wiener index, distance cospectrality and transmission-regular graphs

Aida Abiad, Boris Brimkov, Aysel Erey, Lorinda Leshock, Xavier Martínez-Rivera, Suil O, Sung Yell Song, Jason Williford

Research output: Contribution to journalArticleAcademicpeer-review

20 Citations (Scopus)

Abstract

In this paper, we investigate various algebraic and graph theoretic properties of the distance matrix of a graph. Two graphs are D-cospectral if their distance matrices have the same spectrum. We construct infinite pairs of D-cospectral graphs with different diameter and different Wiener index. A graph is k-transmission-regular if its distance matrix has constant row sum equal to k. We establish tight upper and lower bounds for the row sum of a k-transmission-regular graph in terms of the number of vertices of the graph. Finally, we determine the Wiener index and its complexity for linear k-trees, and obtain a closed form for the Wiener index of block-clique graphs in terms of the Laplacian eigenvalues of the graph. The latter leads to a generalization of a result for trees which was proved independently by Mohar and Merris.

Original languageEnglish
Pages (from-to)1-10
Number of pages10
JournalDiscrete Applied Mathematics
Volume230
DOIs
Publication statusPublished - 30 Oct 2017
Externally publishedYes

Keywords

  • Diameter
  • Distance cospectral graphs
  • Distance matrix
  • Laplacian matrix
  • Transmission-regular
  • Wiener index

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