### Abstract

A universal adjacency matrix of a graph G with adjacency matrix A is any matrix of the form U=αA+βI+γJ+δD with α≠0, where I is the identity matrix, J is the all-ones matrix and D is the diagonal matrix with the vertex degrees. In the case that G is the disjoint union of regular graphs, we present an expression for the characteristic polynomials of the various universal adjacency matrices in terms of the characteristic polynomials of the adjacency matrices of the components. As a consequence we obtain a formula for the characteristic polynomial of the Seidel matrix of G, and the signless Laplacian of the complement of G (i.e. the join of regular graphs). The main tool is a simple but useful lemma on equitable matrix partitions. With this note we also want to propagate this technique.

Original language | English |
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Pages (from-to) | 244-248 |

Journal | Linear Algebra and its Applications |

Volume | 606 |

DOIs | |

Publication status | Published - Dec 2020 |

### Keywords

- Characteristic polynomial
- Graph spectrum
- Laplacian
- Seidel matrix
- Signless Laplacian
- Universal adjacency matrix

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## Cite this

Haemers, W. H., & Oboudi, M. R. (2020). Universal spectra of the disjoint union of regular graphs.

*Linear Algebra and its Applications*,*606*, 244-248. https://doi.org/10.1016/j.laa.2020.07.033