Some results on {K2,C2i+1:i≥1}-factor in a graph

An $\{A,B,C,\dots \}$-factor of a graph $G$ is defined to be a spanning subgraph of $G$ such that each component of which is isomorphic to one of $\{A,B,C,\dots \}$. Let $A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )A\left(G\right)$, where $D\left(G\right)$ denotes the diagonal matrix of vertex degrees of $G$ and $A\left(G\right)$ denotes the adjacency matrix of $G$. The largest eigenvalue of $A_{\alpha }\left(G\right)$ is called the $A_{\alpha }$-spectral radius of $G$. In this paper, we explore the connections between the eigenvalues and the existence of a $\{K_2,C_{2i+1}:i\ge 1\}$-factor in a graph. We derive a tight sufficient condition involving the $A_{\alpha }$-spectral radius to ensure the existence of a $\{K_2,C_{2i+1}:i\ge 1\}$-factor in a graph, which generalizes the result on $\alpha =0$ obtained by Chen, Lv and Li [8]. Moreover, we present a tight distance signless Laplacian spectral radius condition for the existence of a $\{K_2,C_{2i+1}:i\ge 1\}$-factor in a graph.