### Abstract

A mixed extension of a graph $G$ is a graph $H$ obtained from $G$ by replacing each vertex of $G$ by a clique or a coclique, where vertices of $H$ coming from different vertices of $G$ are adjacent if and only if the original vertices are adjacent in $G$. If $G$ has no more than three vertices, $H$ has all but at most three adjacency eigenvalues equal to $0$ or $-1$. In this paper we consider the converse problem, and determine the class $\cal G$ of all graphs with at most three eigenvalues unequal to $0$ and $-1$. Ignoring isolated vertices, we find that $\cal G$ consists of all mixed extensions of graphs on at most three vertices together with some particular mixed extensions of the paths $P_4$ and $P_5$.

Original language | English |
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Pages (from-to) | 2760-2764 |

Journal | Discrete Mathematics |

Volume | 342 |

Issue number | 10 |

Early online date | Feb 2018 |

DOIs | |

Publication status | Published - Oct 2019 |

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### Keywords

- graph spectrum
- spectral characterization