The existence of even factors based on spectral conditions of graphs

TL;DR

This paper establishes spectral conditions involving the signless Laplacian and distance spectral radii of a graph that guarantee the existence of an even factor, a spanning subgraph where each vertex has an even degree.

Contribution

It provides new spectral bounds that ensure the existence of even factors in graphs, extending previous degree-based conditions.

Findings

Lower bound on signless Laplacian spectral radius for even factors

Upper bound on distance spectral radius for even factors

Spectral conditions guarantee the existence of even factors

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 $\delta(G)\geq 2$ is a trivial necessary condition for a graph to have an even factor, where \( \delta(G) \) is the minimum degree of \( G \). In this paper, for a connected graph $G$ with minimum degree $\delta$, 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.