The generating function of whitworth runs (Q800926)

Consider the problem of obtaining, for a graph G, the generating function that gives the number of ways of selecting k vertices such that the subgraph induced by these vertices has exactly h edges. When G is a cycle, this is the Whitworth bracelet problem. \textit{F. K. Hwang} [Proc. Am. Math. Soc. 83, 215-219 (1981; Zbl 0478.05005)] has studied the case when G consists of m components that are cycles of possibly different lengths. This paper includes further technical work on these generating functions. This enables the cases to be studied in which G has 2 components that are one cycle and one path, or two paths.