Spectral and sharp sufficient conditions for a {K1,1,K1,2,Cm : m ≥ 3}-factor

Let [Formula: see text] be a graph and let [Formula: see text] be an integer. A [Formula: see text]-factor of a graph is a spanning subgraph whose each component is an element of [Formula: see text]. In this paper, through the typical spectral techniques, we obtain three sufficient conditions to guarantee that a graph contains a [Formula: see text]-factor. These three spectral sufficient conditions include the following: The size condition, the spectral radius condition and the distance spectral radius condition. Furthermore, by constructing extremal graphs, we show that the bounds are best possible.