### Abstract

The distance matrix of a graph G is the matrix containing the pairwise distances between vertices. The distance eigenvalues of G are the eigenvalues of its distance matrix and they form the distance spectrum of G. We determine the distance spectra of double odd graphs and Doob graphs, completing the determination of distance spectra of distance regular graphs having exactly one positive distance eigenvalue. We characterize strongly regular graphs having more positive than negative distance eigenvalues. We give examples of graphs with few distinct distance eigenvalues but lacking regularity properties. We also determine the determinant and inertia of the distance matrices of lollipop and barbell graphs.

Original language | English |
---|---|

Pages (from-to) | 66-87 |

Number of pages | 22 |

Journal | Linear Algebra and Its Applications |

Volume | 497 |

DOIs | |

Publication status | Published - 15 May 2016 |

### Keywords

- Distance matrix
- Eigenvalue
- Distance regular graph
- Kneser graph
- Double odd graph
- Doob graph
- Lollipop graph
- Barbell graph
- Distance spectrum
- Strongly regular graph
- Optimistic graph
- Determinant
- Inertia
- Graph
- MATRIX

## Cite this

*Linear Algebra and Its Applications*,

*497*, 66-87. https://doi.org/10.1016/j.laa.2016.02.018