# Relations between (*κ, τ*)-regular sets and star complements

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## Abstract

Let *G* be a finite graph with an eigenvalue µ of multiplicity *m*. A set *X* of *m* vertices in *G* is called a star set for µ in *G* if µ is not an eigenvalue of the star complement *G\X* which is the subgraph of *G* induced by vertices not in *X*. A vertex subset of a graph is (*κ, τ*)-regular if it induces a *κ*-regular subgraph and every vertex not in the subset has *τ* neighbors in it. We investigate the graphs having a (*κ, τ*)-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.

## Keywords

eigenvalue star complement non-main eigenvalue Hamiltonian graph## MSC 2010

05C50## Preview

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© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2013