Laplacian Distribution and Domination

TL;DR

This paper explores relationships between Laplacian eigenvalues and domination number in graphs, extending known bounds and revealing new inequalities that connect spectral properties with graph domination and structure.

Contribution

It extends existing bounds on Laplacian eigenvalues and domination number, providing new inequalities and insights into spectral parameters' relation to graph domination.

Findings

For isolate-free graphs, mma(G)  m_G[2,n]

mbda(G)/m_G[0,1)  O(g n)

For trees, mbda(T)  2  mbda(G)

Abstract

Let $m_G(I)$ denote the number of Laplacian eigenvalues of a graph $G$ in an interval $I$, and let $\gamma(G)$ denote its domination number. We extend the recent result $m_G[0,1) \leq \gamma(G)$, and show that isolate-free graphs also satisfy $\gamma(G) \leq m_G[2,n]$. In pursuit of better understanding Laplacian eigenvalue distribution, we find applications for these inequalities. We relate these spectral parameters with the approximability of $\gamma(G)$, showing that $\frac{\gamma(G)}{m_G[0,1)} \not\in O(\log n)$. However, $\gamma(G) \leq m_G[2, n] \leq (c + 1) \gamma(G)$ for $c$-cyclic graphs, $c \geq 1$. For trees $T$, $\gamma(T) \leq m_T[2, n] \leq 2 \gamma(G)$.