On a class of polynomials associated with the paths in a graph and its application to minimum nodes disjoint path coverings of graphs

Let G be a graph. With every path α of G let us associate a weight w α With every spanning subgraph C of G consisting of paths α 1 , α 2 , … , α k , let us associate the weight w ( C ) = ∏ i = 1 k w α i . The path polynomial of G is ∑ w ( C ) , where the summation is taken over all the spanning subgraphs of G whose components are paths. Some basic properties of these polynomials are given. The polynomials are then used to obtain results about the minimum number of node disjoint path coverings in graphs.