On spectral conditions for fractional $k$-extendable graphs

Abstract

A fractional matching of a graph $G$ is a function $h:E(G)\to [0,1]$ such that ${\sum }_{e\in E_G(v)}h(e)\le 1$ 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.

arXiv comment15 pages