Abstract
A set of vertices S ⊆ V ( G ) is ( k , τ ) -regular if it induces a k-regular subgraph of G such that | N G ( v ) ∩ S | = τ ∀ v ∉ S . Note that a connected graph with more than one edge has a perfect matching if and only if its line graph has a ( 0 , 2 ) -regular set. In this paper, some spectral results on the adjacency matrix of graphs with ( k , τ ) -regular sets are presented. Relations between the combinatorial structure of a p-regular graph with a ( k , τ ) -regular set and the eigenspace corresponding to each eigenvalue λ ∉ { p , k - τ } are deduced. Finally, additional results on the effects of Seidel switching (with respect to a bipartition induced by S) of regular graphs are also introduced.
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