Spectral conditions for component factors in graphs involving minimum degree

Abstract

A spanning subgraph $H$ of a graph $G$ is called an $\{P_2,C_3,P_5,T\left(3\right)\}$-factor if every component of $H$ is isomorphic to an element of $\{P_2,C_3,P_5,T\left(3\right)\}$, where $T\left(3\right)$ is one special family of tree. Let $A\left(G\right)$ and $Q\left(G\right)$ denote the adjacency matrix and the signless Laplacian matrix of $G$, respectively. The largest eigenvalues of $A\left(G\right)$ and $Q\left(G\right)$, denoted by $\rho \left(G\right)$ and $q\left(G\right)$, are called the adjacency spectral radius and the signless Laplacian spectral radius of $G$, respectively. In this paper, we first present a sufficient condition to guarantee that a connected graph $G$ with minimum degree $\delta$ contains a $\{P_2,C_3,P_5,T\left(3\right)\}$-factor with respect to its adjacency spectral radius, then we claim a sufficient condition to ensure that a connected graph $G$ with minimum degree $\delta$ has a $\{P_2,C_3,P_5,T\left(3\right)\}$-factor via its signless Laplacian spectral radius.