Spectral radius and fractional matchings in graphs

TL;DR

This paper establishes bounds on the fractional matching number of a graph based on spectral properties, specifically relating the largest eigenvalue and minimum degree, and characterizes the conditions for equality.

Contribution

It introduces new spectral bounds for the fractional matching number and characterizes cases of equality, advancing understanding of graph matchings via eigenvalues.

Findings

If (G) < d1 + 2k/(n-k), then '*(G) > (n-k)/2

Derived a lower bound '*(G)  nd^2 / ((G)^2 + d^2)

Characterized when the bounds are tight

Abstract

A {\it fractional matching} of a graph $G$ is a function $f$ giving each edge a number in $[0,1]$ so that $\sum_{e \in \Gamma(v)} f(e) \le 1$ for each $v\in V(G)$, where $\Gamma(v)$ is the set of edges incident to $v$. The {\it fractional matching number} of $G$, written $\alpha'_*(G)$, is the maximum of $\sum_{e \in E(G)} f(e)$ over all fractional matchings $f$. Let $G$ be an $n$-vertex connected graph with minimum degree $d$, let $\lambda_1(G)$ be the largest eigenvalue of $G$, and let $k$ be a positive integer less than $n$. In this paper, we prove that if $\lambda_1(G) < d\sqrt{1+\frac{2k}{n-k}}$, then $\alpha'_*(G) > \frac{n-k}{2}$. As a result, we prove $\alpha'_*(G) \ge \frac{nd^2}{\lambda_1(G)^2 + d^2}$, we characterize when equality holds in the bound.