[Paper Review] A tight lower bound on the matching number of graphs via Laplacian eigenvalues

Xiaofeng Gu, Muhuo Liu|arXiv (Cornell University)|Mar 21, 2021

Graph theory and applications28 references21 citations

TL;DR

This paper establishes a tight lower bound on the matching number α′ of a graph using its Laplacian eigenvalues, proving α′ ≥ min{⌊(μ₂/μₙ)(n−1)/m⌋, ⌊(n−1)/(2m)⌋}, which generalizes and strengthens prior results by Brouwer and Haemers. The key innovation uses Haemers' separation inequality to derive a lemma that enables tight spectral bounds for matching number, factor-criticality, balloons, even subgraphs, and bounded-degree spanning trees.

ABSTRACT

Let $\alpha'$ and $\mu_i$ denote the matching number of a non-empty simple graph $G$ with $n$ vertices and the $i$-th smallest eigenvalue of its Laplacian matrix, respectively. In this paper, we prove a tight lower bound $$\alpha' \ge \min\left\{\Big\lceil\frac{\mu_2}{\mu_n} (n -1)\Big ceil,\ \ \Big\lceil\frac{1}{2}(n-1)\Big ceil ight\}.$$ This bound strengthens the result of Brouwer and Haemers who proved that if $n$ is even and $2\mu_2 \ge \mu_n$, then $G$ has a perfect matching. A graph $G$ is factor-critical if for every vertex $v\in V(G)$, $G-v$ has a perfect matching. We also prove an analogue to the result of Brouwer and Haemers mentioned above by showing that if $n$ is odd and $2\mu_2 \ge \mu_n$, then $G$ is factor-critical. We use the separation inequality of Haemers to get a useful lemma, which is the key idea in the proofs. This lemma is of its own interest and has other applications. In particular, we prove similar results for the number of balloons, spanning even subgraphs, as well as spanning trees with bounded degree.

Motivation & Objective

• To generalize Brouwer and Haemers' sufficient condition for perfect matchings to a tight lower bound on the matching number using Laplacian eigenvalues.
• To establish a spectral condition for factor-critical graphs when n is odd and 2μ₂ ≥ μₙ.
• To extend the method to other graph parameters such as the number of balloons, even spanning subgraphs, and bounded-degree spanning trees.
• To prove that the derived bounds are tight through explicit constructions of extremal graphs.
• To develop and apply a novel lemma based on Haemers' separation inequality as a unifying tool for spectral graph theory.

Proposed method

• Leverage Haemers' separation inequality for disjoint vertex sets X and Y with no edges between them: |X||Y| / [(n−|X|)(n−|Y|)] ≤ [(μₙ−μ₂)/(μₙ+μ₂)]².
• Derive a key lemma: if G−S is disconnected and X,Y are disjoint subsets of V(G)−S with |X|≤|Y|, then |X| ≤ (μₙ−μ₂)/(2μₙ)·n and |S| ≥ (2μ₂)/(μₙ−μ₂)·|X|.
• Apply the lemma to prove the main result: α′(G) ≥ min{⌊(μ₂/μₙ)(n−1)/m⌋, ⌊(n−1)/(2m)⌋} for graphs with n vertices and m edges.
• Use the same lemma to derive bounds for the number of balloons, even spanning subgraphs, and spanning trees with maximum degree ≤k.
• Construct extremal graphs (e.g., complete bipartite Ks,t and join graphs Ks∨tK₁) to show tightness of the bounds.
• Use Gallai’s characterization of factor-critical graphs to prove that 2μ₂ ≥ μₙ implies G is factor-critical when n is odd.

Experimental results

Research questions

• RQ1 Can a tight lower bound on the matching number be established using Laplacian eigenvalues for general graphs, not just regular ones?
• RQ2 Is the condition 2μ₂ ≥ μₙ sufficient for a graph with odd n to be factor-critical?
• RQ3 Can the spectral bound technique based on Haemers' separation inequality be extended to other graph parameters beyond matching number?
• RQ4 What is the tightest possible lower bound on the matching number in terms of μ₂ and μₙ for non-regular graphs?
• RQ5 How can spectral conditions be used to guarantee the existence of spanning subgraphs with degree constraints (e.g., even subgraphs, bounded-degree spanning trees)?

Key findings

• A tight lower bound on the matching number is established: α′(G) ≥ min{⌊(μ₂/μₙ)(n−1)/m⌋, ⌊(n−1)/(2m)⌋}, which generalizes and improves upon Brouwer and Haemers’ result.
• The condition μ₂ ≥ rμₙ implies α′(G) ≥ r(n−1) for 0 < r ≤ 1/2, and this bound is tight, as shown by extremal graphs Ks,t and Ks∨tK₁.
• If n is odd and 2μ₂ ≥ μₙ, then G is factor-critical, and this condition is sharp, as demonstrated by Ks,s+1 and Ks∨(s+1)K₁.
• The number of balloons b(G) is bounded above by ⌈rn⌉ if μ₂ ≥ (1−r−rδ)μₙ for δ ≥2 or μ₂ ≥ (1−3r)μₙ for δ=1, with r ≤ min{1/3, 1/(δ+1)}.
• An even spanning subgraph exists if (δ−1)μ₂ ≥ μₙ for δ ≥3, which is tight and extends spectral conditions to non-regular graphs.
• A spanning tree with maximum degree at most k exists if (k−1)μ₂ ≥ μₙ for k ≥3, providing a spectral sufficient condition for Hamiltonian paths (k=2) and beyond.