On the Wiener index, distance cospectrality and transmission regular graphs

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

Research output: Contribution to journalArticleAcademicpeer-review

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. (C) 2017 Elsevier B.V. All rights reserved.

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

Keywords

  • Distance matrix
  • Distance cospectral graphs
  • Diameters
  • Wiener index
  • Laplacian matrix
  • Transmission-regular
  • BALANCED GRAPHS
  • MATRIX
  • NUMBER
  • LARGEST EIGENVALUE
  • WIDTH
  • FORBIDDEN MINORS
  • TREES
  • SPECTRUM
  • VERTICES
  • Diameter

Cite this

Abiad Monge, A., Brimkov, B., Erey, A., Leshock, L., Martínez-Rivera, X., O, S., Song, S-Y., & Williford, J. (2017). On the Wiener index, distance cospectrality and transmission regular graphs. Discrete Applied Mathematics, 230, 1-10. https://doi.org/10.1016/j.dam.2017.07.010