[Paper Review] Signless Laplacian spectral radius and matching in graphs

Chang Liu, Yingui Pan+1 more|arXiv (Cornell University)|Jul 8, 2020

Graph theory and applications20 references4 citations

TL;DR

This paper establishes a sharp spectral condition for perfect matchings in even-order connected graphs using the signless Laplacian spectral radius. It proves that if the largest eigenvalue of the signless Laplacian matrix exceeds a specific threshold—defined by the largest root of a cubic equation for n ≥ 10 or n = 4, and explicit values for n = 6 and n = 8—then the graph must have a perfect matching, with the bound being tight due to extremal graphs like $K_{n-3}\vee K_1\vee \overline{K_2}$.

ABSTRACT

The signless Laplacian matrix of a graph $G$ is given by $Q(G)=D(G)+A(G)$, where $D(G)$ is a diagonal matrix of vertex degrees and $A(G)$ is the adjacency matrix. The largest eigenvalue of $Q(G)$ is called the signless Laplacian spectral radius, denoted by $q_1=q_1(G)$. In this paper, some properties between the signless Laplacian spectral radius and perfect matching in graphs are establish. Let $r(n)$ be the largest root of equation $x^3-(3n-7)x^2+n(2n-7)x-2(n^2-7n+12)=0$. We show that $G$ has a perfect matching for $n=4$ or $n\geq10$, if $q_1(G)>r(n)$, and for $n=6$ or $n=8$, if $q_1(G)>4+2\sqrt{3}$ or $q_1(G)>6+2\sqrt{6}$ respectively, where $n$ is a positive even integer number. Moreover, there exists graphs $K_{n-3}\vee K_1 \vee \overline{K_2}$ such that $q_1(K_{n-3}\vee K_1 \vee \overline{K_2})=r(n)$ if $n\geq4$, a graph $K_2\vee\overline{K_4}$ such that $q_1(K_2\vee\overline{K_4})=4+2\sqrt{3}$ and a graph $K_3\vee\overline{K_5}$ such that $q_1(K_3\vee\overline{K_5})=6+2\sqrt{6}$. These graphs all have no prefect matching.

Motivation & Objective

• To determine a sufficient spectral condition based on the signless Laplacian spectral radius for the existence of perfect matchings in even-order connected graphs.
• To extend prior work on spectral conditions for matchings by focusing on the signless Laplacian matrix rather than adjacency or Laplacian matrices.
• To identify extremal graphs that achieve the threshold spectral radius without perfect matchings, proving the sharpness of the bound.
• To generalize results from previous works on edge counts and spectral radii by introducing a unified spectral threshold.

Proposed method

• Define the signless Laplacian matrix $ Q(G) = D(G) + A(G) $, where $ D(G) $ is the degree matrix and $ A(G) $ the adjacency matrix.
• Introduce $ r(n) $, the largest root of the cubic equation $ x^3 - (3n-7)x^2 + n(2n-7)x - 2(n^2 - 7n + 12) = 0 $, as a spectral threshold for $ n \geq 10 $ or $ n = 4 $.
• For $ n = 6 $ and $ n = 8 $, use explicit values $ 4 + 2\sqrt{3} $ and $ 6 + 2\sqrt{6} $, respectively, as thresholds.
• Apply equitable partitioning of the signless Laplacian matrix to compute the spectral radius of extremal graphs like $ K_{n-3} \vee K_1 \vee \overline{K_2} $, $ K_2 \vee \overline{K_4} $, and $ K_3 \vee \overline{K_5} $.
• Use the interlacing property and nonnegative matrix theory to compare spectral radii and establish the sharpness of the bound.
• Verify that the spectral radius of the extremal graphs equals the threshold, confirming that graphs meeting the bound may still lack perfect matchings.

Experimental results

Research questions

• RQ1 What spectral condition on the signless Laplacian spectral radius guarantees the existence of a perfect matching in an even-order connected graph?
• RQ2 How does the threshold spectral radius depend on the number of vertices, particularly for small even $ n $?
• RQ3 Are there extremal graphs that achieve the spectral threshold but do not have perfect matchings, thus proving the sharpness of the bound?
• RQ4 How does the spectral radius of the join graph $ K_{n-3} \vee K_1 \vee \overline{K_2} $ relate to the threshold $ r(n) $ for $ n \geq 4 $?
• RQ5 Why do the thresholds for $ n = 6 $ and $ n = 8 $ differ from the general $ r(n) $ formula, and how are they derived?

Key findings

• For even $ n \geq 4 $, if $ q_1(G) > r(n) $ with $ r(n) $ being the largest root of the cubic equation $ x^3 - (3n-7)x^2 + n(2n-7)x - 2(n^2 - 7n + 12) = 0 $, then $ G $ has a perfect matching when $ n \geq 10 $ or $ n = 4 $.
• For $ n = 6 $, the threshold is $ 4 + 2\sqrt{3} \approx 7.464 $, and if $ q_1(G) $ exceeds this value, $ G $ has a perfect matching.
• For $ n = 8 $, the threshold is $ 6 + 2\sqrt{6} \approx 10.899 $, and exceeding it ensures a perfect matching.
• The bound is sharp: there exist graphs like $ K_{n-3} \vee K_1 \vee \overline{K_2} $ with $ q_1(G) = r(n) $ and no perfect matching.
• The spectral radius of $ K_2 \vee \overline{K_4} $ is exactly $ 4 + 2\sqrt{3} $, and of $ K_3 \vee \overline{K_5} $ is exactly $ 6 + 2\sqrt{6} $, confirming the sharpness of the special-case thresholds.
• The extremal graphs $ K_{n-3} \vee K_1 \vee \overline{K_2} $, $ K_2 \vee \overline{K_4} $, and $ K_3 \vee \overline{K_5} $ achieve the spectral threshold and lack perfect matchings, proving the bound cannot be improved.