On spectral conditions for fractional $k$-extendable graphs

TL;DR

This work addresses when a connected graph with minimum degree $\delta$ is fractionally $k$-extendable by establishing spectral and edge-count thresholds. It develops three sufficiency criteria based on the edge count, the distance spectral radius $\mu(G)$, and the signless Laplacian spectral radius $q(G)$, each compared against explicit extremal joins like $K_{2k} \vee (K_{n-2k-1} \cup K_1)$ and related structures. The authors employ the standard $k$-extendability criterion via subgraph obstructions, equitable partitions with quotient matrices, and spectral monotonicity to rule out non-extendable configurations, proving sharp thresholds under various regimes. The results unify and extend prior work by providing concrete extremal graphs and precise spectral-edge conditions that guarantee fractional $k$-extendability, with potential implications for spectral graph theory and combinatorial optimization.

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 $δ$ 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.

Theorems & Definitions (14)

• Theorem 1.1: MR4822576
• Theorem 1.2: MR4600143
• Theorem 1.3: MR4899831
• Theorem 1.4: MR5004976
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1: MR2102027
• Lemma 2.2: MR3589612
• Lemma 2.3: MR932967