On the Wiener Index, Distance Cospectrality and Transmission-regular Graphs

Authors

Aida Abiad, Maastricht University
Boris Brimkov, Rice University
Aysel Erey, University of Denver
Lorinda Leshock, University of Delaware
Xavier Martínez-Rivera, Iowa State University
Suil O, The State University of New York-Korea
Sung-Yell Song, Iowa State University
Jason Williford, University of Wyoming

Publication Date

7-29-2017

Document Type

Article

Organizational Units

Mathematics

Keywords

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

Abstract

In this paper, we investigate various algebraic and graph theoretic properties of the distance matrix of a graph. Two graphs are D" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">D-cospectral if their distance matrices have the same spectrum. We construct infinite pairs of D" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">D-cospectral graphs with different diameter and different Wiener index. A graph is k" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">k-transmission-regular if its distance matrix has constant row sum equal to k" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">k. We establish tight upper and lower bounds for the row sum of a k" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">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" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">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.

Publication Statement

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Recommended Citation

Abiad, Aida, et al. “On the Wiener Index, Distance Cospectrality and Transmission-Regular Graphs.” Discrete Applied Mathematics, vol. 230, 2017, pp. 1–10. doi: 10.1016/j.dam.2017.07.010.