# Article

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Keywords:
eigenvalue; star complement; non-main eigenvalue; Hamiltonian graph
Summary:
Let $G$ be a finite graph with an eigenvalue $\mu$ of multiplicity $m$. A set $X$ of $m$ vertices in $G$ is called a star set for $\mu$ in $G$ if $\mu$ is not an eigenvalue of the star complement $G\setminus X$ which is the subgraph of $G$ induced by vertices not in $X$. A vertex subset of a graph is $(\kappa ,\tau )$-regular if it induces a $\kappa$-regular subgraph and every vertex not in the subset has $\tau$ neighbors in it. We investigate the graphs having a $(\kappa ,\tau )$-regular set which induces a star complement for some eigenvalue. A survey of known results is provided and new properties for these graphs are deduced. Several particular graphs where these properties stand out are presented as examples.
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