Spectral Sufficient Conditions for Graph Factors

TL;DR

This paper uses spectral graph theory to establish bounds on spectral radii that guarantee the existence of certain graph factors, providing optimal conditions and extremal examples.

Contribution

It introduces spectral conditions for the existence of specific graph factors, including bounds on signless Laplacian and distance spectral radii, with proven optimality.

Findings

Lower bounds on size and spectral radius for graph factors

Upper bounds on distance spectral radius for graph factors

Construction of extremal graphs demonstrating bounds' sharpness

Abstract

The $\{K_{1,1}, K_{1,2},C_m: m\geq3\}$-factor of a graph is a spanning subgraph whose each component is an element of $\{K_{1,1}, K_{1,2},C_m: m\geq3\}$. In this paper, through the graph spectral methods, we establish the lower bound of the signless Laplacian spectral radius and the upper bound of the distance spectral radius to determine whether a graph admits a $\{K_2\}$-factor. We get a lower bound on the size (resp. the spectral radius) of $G$ to guarantee that $G$ contains a $\{K_{1,1}, K_{1,2},C_m: m\geq3\}$-factor. Then we determine an upper bound on the distance spectral radius of $G$ to ensure that $G$ has a $\{K_{1,1}, K_{1,2},C_m: m\geq3\}$-factor. Furthermore, by constructing extremal graphs, we show that the above all bounds are best possible.