### Abstract

Properties of graphs that can be characterized by the spectrum of the adjacency matrix of the graph have been studied systematically recently. Motivated by the complexity of these properties, we show that there are such properties for which testing whether a graph has that property can be NP-hard (or belong to other computational complexity classes consisting of even harder problems). In addition, we discuss a possible spectral characterization of some well-known NP-hard properties. In particular, for every integer k≥6 we construct a pair of k-regular cospectral graphs, where one graph is Hamiltonian and the other one not.

Original language | English |
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Pages (from-to) | 526-529 |

Journal | Discrete Applied Mathematics |

Volume | 285 |

DOIs | |

Publication status | Published - 15 Oct 2020 |

### Keywords

- Computational complexity
- Cospectral graphs
- Hamiltonian graph
- Latin square
- NP-hard problem
- Spectral characterization

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## Cite this

Etesami, O., & Haemers, W. H. (2020). On NP-hard graph properties characterized by the spectrum.

*Discrete Applied Mathematics*,*285*, 526-529. https://doi.org/10.1016/j.dam.2020.06.020