Spectral conditions for graphs to contain $k$-factors

TL;DR

This paper establishes spectral radius conditions that ensure large graphs with minimum degree at least k contain a k-factor, providing sharp bounds for such spectral conditions.

Contribution

It introduces a precise spectral radius lower bound that guarantees the existence of a k-factor in graphs with given minimum degree and order.

Findings

Derived a sharp lower bound for spectral radius to ensure k-factors.

Extended spectral conditions to graphs with specified minimum degree and order.

Provided theoretical proof of the spectral condition's sufficiency.

Abstract

Let $G$ be a graph. The spectral radius $\rho(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. For an integer $k\geq1$, a $k$-factor of $G$ is a $k$-regular spanning subgraph of $G$. Assume that $k$ and $n$ are integers satisfying $k\geq2,kn\equiv0~(\mod2)$ and $n\geq\max\left\{k^{2}+6k+7,20k+10\right\}$. Let $G$ be a graph of order $n$ and with minimum degree at least $k$. In this paper, we give a sharp lower bound of $\rho(G)$ to guarantee that $G$ contains a $k$-factor.