Spectral characterization of mixed extensions of small graphs

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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 languageEnglish
Pages (from-to)2760-2764
JournalDiscrete Mathematics
Issue number10
Early online dateFeb 2018
Publication statusPublished - Oct 2019


  • graph spectrum
  • spectral characterization


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