On graph-based network parameters and component factors in networks

\nMany 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\n\n$ \\sigma_k{(G)}=\\underset{X\\subseteq V(G)}\\min\\,{\\{\\underset{x\\in X}{\\mathrm\\Sigma}\\,d_G{(x)}:X\\text{is an independent set of}k\\text{vertices}\\}.}$\n\nIn this article, we give the precise degree sum condition for a graph to have {P2, C3, P5, T (3)}-factor and {K1,1, K1,2, …, K1,k, T (2k + 1)}-factor. We also obtain similar results for {P2, C3, P5, T (3)}-factor avoidable graph and {K1,1, K1,2, …, K1,k, T (2k + 1)}-factor avoidable graph, respectively.\n