Abstract
Many physical structures can conveniently be simulated by networks. To study the properties of the network, we use a graph to simulate the network. A graph H is called an F-factor of a graph G, if H is a spanning subgraph of G and every connected component of H is isomorphic to a graph from the graph set F. An F-factor is also referred as a component factor. The graph-based network parameter degree sum of G is defined by $\sigma_k(G)=\min_{X\subseteq V(G)}\{\sum_{x\in X}d_G(x): X is an independent set of k vertices\}.$ In this article, we give the precise degree sum condition for a graph to have $\{P_2,C_3, P_5, \mathcal{T}（3）\}$-factor and $\{K_{1,1}, K_{1,2},..., K_{1,k},\mathcal{T}(2k+1)\}$-factor. We also obtain similar results for $\{P_2,C_3, P_5, \mathcal{T}（3）\}$-factor avoidable graph and $\{K_{1,1}, K_{1,2},..., K_{1,k},\mathcal{T}(2k+1)\}$-factor avoidable graph, respectively.
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