A bound for the p-domination number of a graph in terms of its eigenvalue multiplicities
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Let G be a connected graph of order n with domination number γ(G). Wang, Yan, Fang, Geng and Tian [Linear Algebra Appl. 607 (2020), 307-318] showed that for any Laplacian eigenvalue λ of G with multiplicity mG(λ), it holds that γ(G)≤n−mG(λ). Using techniques from the theory of star sets, in this work we prove that the same bound holds when λ is an arbitrary adjacency eigenvalue of a non-regular graph, and we characterize the cases of equality. Moreover, we show a result that gives a relationship between start sets and the p-domination number, and we apply it to extend the aforementioned spectral bound to the p-domination number using the adjacency and Laplacian eigenvalue multiplicities.