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arXiv · 1609.06911

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
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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.

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Aida Abiad, Boris Brimkov, Aysel Erey, Lorinda Leshock, Xavier Martínez-Rivera, Suil O, Sung-Yell Song, Jason Williford. 2016-09-22. On the Wiener index, distance cospectrality and transmission regular graphs. https://arxiv.org/abs/1609.06911

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