On spectral conditions for fractional $k$-extendable graphs

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 $δ$ 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.

💡 Research Summary

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The paper investigates sufficient spectral conditions under which a graph G with minimum degree δ is fractional k‑extendable. A fractional matching is a function h: E(G) →