Sufficient conditions for a graph with minimum degree to have a component factor

Let ${\mathcal{T}}_{\frac{k}{r}}$ denote the set of trees $T$ such that $i(T-S)\le \frac{k}{r}\mathrm{\mid }S\mathrm{\mid }$ for any $S\subset V(T)$ and for any $e\in E(T)$ there exists a set $S^{\ast }\subset V(T)$ with $i((T-e)-S^{\ast })>\frac{k}{r}\mathrm{\mid }{S}^{\ast }\mathrm{\mid }$, where $r<k$ are two positive integers. A $\{C_{2i+1},T:1\le i<\frac{r}{k-r},T\in {\mathcal{T}}_{\frac{k}{r}}\}$-factor of a graph $G$ is a spanning subgraph of $G$, in which every component is isomorphic to an element in $\{C_{2i+1},T:1\le i<\frac{r}{k-r},T\in {\mathcal{T}}_{\frac{k}{r}}\}$. Let $A(G)$ and $Q(G)$ denote the adjacency matrix and the signless Laplacian matrix of $G$, respectively. The adjacency spectral radius and the signless Laplacian spectral radius of $G$, denoted by $\rho (G)$ and $q(G)$, are the largest eigenvalues of $A(G)$ and $Q(G)$, respectively. In this paper, we study the connections between the spectral radius and the existence of a $\{C_{2i+1},T:1\le i<\frac{r}{k-r},T\in {\mathcal{T}}_{\frac{k}{r}}\}$-factor in a graph. We first establish a tight sufficient condition involving the adjacency spectral radius to guarantee the existence of a $\{C_{2i+1},T:1\le i<\frac{r}{k-r},T\in {\mathcal{T}}_{\frac{k}{r}}\}$-factor in a graph. Then we propose a tight signless Laplacian spectral radius condition for the existence of a $\{C_{2i+1},T:1\le i<\frac{r}{k-r},T\in {\mathcal{T}}_{\frac{k}{r}}\}$-factor in a graph.