Sufficient conditions for component factors in a graph

Let G be a graph and \(\mathcal {H}\) $\mathcal{H}$ be a set of connected graphs. A spanning subgraph H of G is called an \(\mathcal {H}\) $\mathcal{H}$ –factor if each component of H is isomorphic to a member of \(\mathcal {H}\) $\mathcal{H}$ . In this paper, we first present a lower bound on the size (resp. the spectral radius) of G to guarantee that G has a \(\{P_2,\, C_n: n\ge 3\}\) $\{P_2,C_n:n\ge 3\}$ –factor (or a perfect k–matching for even k) and construct extremal graphs to show all this bounds are best possible. We then provide a lower bound on the signless laplacian spectral radius of G to ensure that G has a \(\{K_{1,j}:1\le j\le k\}\) $\{K_{1,j}:1\le j\le k\}$ –factor, where \(k\ge 2 \) $k\ge 2$ is an integer. Moreover, we also provide some Laplacian eigenvalue (resp. toughness) conditions for the existence of \(\{P_2,\, C_{n}:n\ge 3\}\) $\{P_2,C_n:n\ge 3\}$ –factor, \(P_{\ge 3}\) $P_{\ge 3}$ –factor and \(\{K_{1,j}: 1\le j\le k\}\) $\{K_{1,j}:1\le j\le k\}$ –factor in G, respectively. Some of our results extend or improve the related existing results.