On graph-based network parameters and component factors in networks

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 σₖ(G)=X⊆ V(G)min\,∈ XΣ\,dG(x):Xis an independent set ofkvertices\. In this article, we give the precise degree sum condition for a graph to have { P 2 , C 3 , P 5 , T (3)}-factor and { K 1,1 , K 1,2 , …, K 1, k , T (2 k + 1)}-factor. We also obtain similar results for { P 2 , C 3 , P 5 , T (3)}-factor avoidable graph and { K 1,1 , K 1,2 , …, K 1,k , T (2 k + 1)}-factor avoidable graph, respectively.