### 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 language | English |
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Pages (from-to) | 1-10 |

Number of pages | 10 |

Journal | Discrete Applied Mathematics |

Volume | 230 |

DOIs | |

Publication status | Published - 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

*Discrete Applied Mathematics*,

*230*, 1-10. https://doi.org/10.1016/j.dam.2017.07.010