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On the Wiener index, distance cospectrality and transmission-regular graphs
Abiad, A.,Brimkov, B.,Erey, A.,Leshock, L.,Martinez-Rivera, X.,O, S.,Song, S.Y.,Williford, J. Elsevier BV, North-Holland 2017 Discrete Applied Mathematics Vol.230 No.-
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.