Graph norms and Sidorenko's conjecture

Let H and G be two finite graphs. Define h (H) (G) to be the number of homomorphisms from H to G. The function h (H) (center dot) extends in a natural way to a function from the set of symmetric matrices to a"e such that for A (G) , the adjacency matrix of a graph G, we have h (H) (A (G) ) = h (H) (G). Let m be the number of edges of H. It is easy to see that when H is the cycle of length 2n, then h (H) (center dot)(1/m) is the 2n-th Schatten-von Neumann norm. We investigate a question of Lovasz that asks for a characterization of graphs H for which the function h (H) (center dot)(1/m) is a norm. We prove that h (H) (center dot)(1/m) is a norm if and only if a Holder type inequality holds for H. We use this inequality to prove both positive and negative results, showing that h (H) (center dot)(1/m) is a norm for certain classes of graphs, and giving some necessary conditions on the structure of H when h (H) (center dot)(1/m) is a norm. As an application we use the inequality to verify a conjecture of Sidorenko for certain graphs including hypercubes. In fact, for such graphs we can prove statements that are much stronger than the assertion of Sidorenko's conjecture. We also investigate the h (H) (center dot)(1/m) norms from a Banach space theoretic point of view, determining their moduli of smoothness and convexity. This generalizes the previously known result for the 2n-th Schatten-von Neumann norms.