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arXiv · 1810.12615

On the spectral characterization of mixed extensions of $P_3$

Willem H. Haemers·Sezer Sorgun·Hatice Topcu
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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, whilst two vertices in $H$ corresponding to distinct vertices $x$ and $y$ of $G$ are adjacent whenever $x$ and $y$ are adjacent in $G$. If $G$ is the path $P_3$, then $H$ has at most three adjacency eigenvalues unequal to $0$ and $-1$. Recently, the first author classified the graphs with the mentioned eigenvalue property. Using this classification we investigate mixed extension of $P_3$ on being determined by the adjacency spectrum. We present several cospectral families, and with the help of a computer we find all graphs on at most $25$ vertices that are cospectral with a mixed extension of $P_3$.

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Willem H. Haemers, Sezer Sorgun, Hatice Topcu. 2018-10-30. On the spectral characterization of mixed extensions of $P_3$. https://arxiv.org/abs/1810.12615

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