Laplacian Spectrum and Domination in Trees

TL;DR

This paper investigates the relationship between the Laplacian spectrum and domination number in trees, establishing a tight upper bound on their ratio and exploring how structural constraints affect this bound.

Contribution

It proves that the ratio of domination number to Laplacian eigenvalues in trees is bounded by 4/3, and refines this bound for trees with specific degree conditions.

Findings

The ratio $rac{ ext{domination number}}{ ext{Laplacian eigenvalues}}$ in trees is less than 4/3.

The bound of 4/3 is tight, as it is approached by an infinite family of trees.

Stronger bounds are obtained for trees with minimum degree constraints, improving the general ratio bound.

Abstract

For a finite simple undirected graph $G$, let $\gamma(G)$ denote the size of a smallest dominating set of $G$ and $\mu(G)$ denote the number of eigenvalues of the Laplacian matrix of $G$ in the interval $[0,1)$, counting multiplicities. Hedetniemi, Jacobs and Trevisan [Eur. J. Comb. 2016] showed that for any graph $G$, $\mu(G) \leqslant \gamma(G)$. Cardoso, Jacobs and Trevisan [Graphs Combin. 2017] asks whether the ratio $\gamma(T)/\mu(T)$ is bounded by a constant for all trees $T$. We answer this question by showing that this ratio is less than $4/3$ for every tree. We establish the optimality of this bound by constructing an infinite family of trees where this ratio approaches $4/3$. We also improve this upper bound for trees in which all the vertices other than leaves and their parents have degree at least $k$, for every $k \geqslant 3$. We show that, for such trees $T$, $\gamma(T)/\mu(T) < 1 + 1/((k-2)(k+1))$.