I am trying to help someone with some homework problems. He is trying very hard, but is stuck on these two. Any help would be greatly appreciated.
1. Determine the number of cycles of length 4 in the complete bipartite graph K_{(m,n)}
2. Let F_{k} denote the number of faces of G (G is a simple connected planar graph with k sides
( For example, every face of K_{2,3} has four sides, so F_4 = 3 for this graph.)
Prove that: 3F_3 + 4F_4 + 5F_5 + ...= 2E.
(E is the number of edges in G)