  Let $H$ and $G$ be two finite graphs. Define $h_H(G)$ to be the number of
homomorphisms from $H$ to $G$. The function $h_H(\cdot)$ extends in a natural
way to a function from the set of symmetric matrices to $\mathbb{R}$ 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 $h_H(\cdot)^{1/m}$ is the $2n$-th Schatten-von Neumann
norm. We investigate a question of Lov\'{a}sz that asks for a characterization
of graphs $H$ for which the function $h_H(\cdot)^{1/m}$ is a norm.
  We prove that $h_H(\cdot)^{1/m}$ is a norm if and only if a H\"{o}lder type
inequality holds for $H$. We use this inequality to prove both positive and
negative results, showing that $h_H(\cdot)^{1/m}$ is a norm for certain classes
of graphs, and giving some necessary conditions on the structure of $H$ when
$h_H(\cdot)^{1/m}$ 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 $h_H(\cdot)^{1/m}$ 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.
