Geodesic
Relations between $(\kappa,\tau)$-regular sets and star complements
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 1, pp. 73-90 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.1007/s10587-013-0005-5
Classification : 05C38, 05C50
Keywords: eigenvalue; star complement; non-main eigenvalue; Hamiltonian graph 
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     title = {Relations between $(\kappa,\tau)$-regular sets and star complements},
     journal = {Czechoslovak Mathematical Journal},
     pages = {73--90},
     year = {2013},
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Anđelić, Milica; Cardoso, Domingos M.; Simić, Slobodan K. Relations between $(\kappa,\tau)$-regular sets and star complements. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 1, pp. 73-90. doi: 10.1007/s10587-013-0005-5

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