A bound for the $p$-domination number of a graph in terms of its eigenvalue multiplicities

TL;DR

This paper extends a spectral bound on the domination number of graphs to the p-domination number, using adjacency and Laplacian eigenvalue multiplicities, and characterizes cases of equality.

Contribution

It generalizes a known bound from Laplacian eigenvalues to arbitrary adjacency eigenvalues for non-regular graphs and relates star sets to p-domination.

Findings

The bound holds for arbitrary adjacency eigenvalues in non-regular graphs.

Characterization of equality cases for the spectral bound.

Extension of the spectral bound to p-domination number.

Abstract

Let $G$ be a connected graph of order $n$ with domination number $\gamma(G)$. Wang, Yan, Fang, Geng and Tian [Linear Algebra Appl. 607 (2020), 307-318] showed that for any Laplacian eigenvalue $\lambda$ of $G$ with multiplicity $m_G(\lambda)$, it holds that $\gamma(G)\leq n-m_G(\lambda)$. Using techniques from the theory of star sets, in this work we prove that the same bound holds when $\lambda$ 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.