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

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. [ABSTRACT FROM AUTHOR]