Laplacian Spectrum and Domination in Trees

TL;DR

The paper resolves whether the domination number $\gamma(T)$ of a tree can be controlled by the number $\mu(T)$ of Laplacian eigenvalues in $[0,1)$, proving $1 \leq \frac{\gamma(T)}{\mu(T)} < \frac{4}{3}$ for every tree $T$, and showing this bound is tight via an infinite family where the ratio approaches $\frac{4}{3}$. The authors introduce a Sylvester-based inertia framework and two algorithms: a standard inertia-based method and a new dominating-set algorithm that runs in tandem, yielding a constructive bound $|D| \le \mu(T) + \frac{1}{3}(p(T)-1)$. They further tighten the ratio for trees with high-degree deep vertices, obtaining $\frac{\gamma(T)}{\mu(T)} < 1 + \frac{1}{(k-2)(k+1)}$ when all deep vertices have degree at least $k$, $k\ge 3$, and demonstrate the tightness of related bounds on the Laplacian spectrum distribution via a caterpillar family. The results advance understanding of how spectral properties constrain domination in trees and inform broader spectral–combinatorial connections in graphs.

Abstract

For a finite simple undirected graph $G$, let $γ(G)$ denote the size of a smallest dominating set of $G$ and $μ(G)$ 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 $G$, $μ(G) \leqslant γ(G)$. Cardoso, Jacobs and Trevisan [Graphs Combin. 2017] asks whether the ratio $γ(T)/μ(T)$ 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 $T$, $γ(T)/μ(T) < 1 + 1/((k-2)(k+1))$.

Figures (4)

• Figure 1: Four subcases when $|F|=3$ and all feeders of $v$ are pseudo-positive in the proof of Lemma \ref{['4by3domination']}
• Figure 2: 11 vertex tree with $\gamma=5$ and $\mu=4$. Inertia of each vertex is indicated.
• Figure 3: Tree with inertia of vertices. The penultimate vertices are indicated by dark circles.
• Figure 4: $T_3$ with inertia of vertices when rooted at $v_9$.

Theorems & Definitions (28)

• Lemma 1: hedetniemi2016domination
• Theorem 2
• Definition 1
• Theorem 3
• Lemma 4: Sylvester's Law of Inertia horn2012matrix
• Theorem 5: jacobs2011locating
• Definition 2
• Corollary 6
• Lemma 7
• proof