On spectral conditions for fractional $k$-extendable graphs

TL;DR

This paper establishes new spectral conditions involving the distance spectral radius and signless Laplacian spectral radius that guarantee a graph's fractional $k$-extendability, a property related to fractional matchings.

Contribution

The paper introduces novel spectral criteria based on spectral radii to ensure fractional $k$-extendability in graphs with given minimum degree.

Findings

Spectral conditions guarantee fractional $k$-extendability.

Distance spectral radius bounds imply fractional $k$-extendability.

Signless Laplacian spectral radius bounds imply fractional $k$-extendability.

Abstract

A fractional matching of a graph $G$ is a function $h: E(G) \to [0,1]$ such that $\sum_{e \in E_G(v)} h(e) \leq 1$ for every vertex $v \in V(G)$, where $E_G(v)$ is the set of edges incident to $v$. If $\sum_{e \in E_G(v)} h(e) = 1$ for all $v$, then $h$ is a fractional perfect matching. A graph $G$ is fractional $k$-extendable if it has a matching of size $k$ and every $k$-matching $M$ in $G$ is contained in a fractional perfect matching $h$ such that $h(e)=1$ for every $e \in M$. In this paper, we establish new sufficient conditions for a graph with minimum degree $\delta$ to be fractional $k$-extendable. Our main results provide spectral guarantees for this property based on the distance spectral radius and the signless Laplacian spectral radius.