Parity theorems for paths and cycles in graphs

We extend an elegant proof technique of A. G. Thomason, and deduce several parity theorems for paths and cycles in graphs. For example, a graph in which each vertex is of even degree has an even number of paths if and only if it is of even order, and a graph in which each vertex is of odd degree has an even number of paths if and only if its order is a multiple of four. Our results have implications for generalized friendship graphs and their conjectured nonexistence.