The existence of even factors based on spectral conditions of graphs

TL;DR

The paper establishes spectral-sufficient conditions for the existence of an even factor in connected graphs with minimum degree $\delta$. By deriving explicit bounds on the signless Laplacian spectral radius $\rho_Q(G)$ (lower bound) and the distance spectral radius $\rho_\mathcal{D}(G)$ (upper bound) relative to the extremal join-graph $G_* = K_{\delta} \lor (K_{n-2\delta+1} \cup (\delta-1)K_1)$, the authors show that an even factor exists unless $G$ is isomorphic to $G_*$. The proofs combine equitable-partition techniques and quotient matrices with case analyses on the parameter $s=|S|$ from a structural obstruction, establishing tight thresholds that depend on $n$ and $\delta$. These results extend combinatorial conditions for even factors into the spectral domain, providing practical criteria via eigenvalue calculations and a precise characterization of the extremal graphs.

Abstract

Let $G=(V(G),E(G)) $ be a graph with vertex set $V(G)$ and edge set $E(G)$. An even factor of $G$ is a spanning subgraph $F$ such that every vertex in $F$ has a nonzero even degree. Note that $δ(G)\geq 2$ is a trivial necessary condition for a graph to have an even factor, where \( δ(G) \) is the minimum degree of \( G \). In this paper, for a connected graph $G$ with minimum degree $δ$, we establish a lower bound on the signless Laplacian spectral radius of $G$ and an upper bound on the distance spectral radius of $G$ such that $G$ contains an even factor.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: 13
• Lemma 2.2: 5
• Lemma 2.3: 19
• Lemma 2.4: 19
• Lemma 2.5: 17
• Lemma 2.6: 21
• Lemma 2.7
• Lemma 2.8: 11