Signless Laplacian spectral conditions for even factors in graphs
Lu Li, Hechao Liu, Hongbo Hua, Zenan Du
TL;DR
The paper proves a sharp signless Laplacian spectral radius condition guaranteeing the existence of an even factor in connected graphs of even order with minimum degree $\delta\ge 2$, namely $q(G) \ge q(K_\delta \vee (K_{n-2\delta+1} \cup (\delta-1)K_1))$, with equality only for the extremal join $K_\delta \vee (K_{n-2\delta+1} \cup (\delta-1)K_1)$. The authors develop a contradiction framework, reducing to an extremal join graph and leveraging equitable partitions and Perron-vector properties to compare spectral radii. The proof tackles three cases based on the size of the obstruction set and culminates in a spectral-bound violation unless an even factor exists, thus establishing the result. This work extends spectral-radius criteria for spanning subgraph existence to the realm of even factors and provides a concrete extremal graph that characterizes the threshold.
Abstract
A spanning subgraph $F$ of a graph $G$ is defined as an even factor of $G$, if the degree $d_F(v)=2k, k\in\mathbb{N}^+$ for every vertex $v\in V(G)$. This note establishes a sufficient condition to ensure that a connected graph $G$ of even order with the minimum degree $δ$ contains an even factor based on the signless Laplacian spectral radius.