Weakly Norming Graphs are Edge-Transitive

Abstract

Let ℋ be the class of bounded measurable symmetric functions on [0, 1]². For a function h ∈ ℋ and a graph G with vertex set [v₁,⌦,v_n} and edge set E(G), define

$${t_G}(h) = \int \cdots \int {\prod\limits_{{\rm{\{ }}{v_i}{\rm{,}}{v_j}{\rm{\} }} \in E(G)} {h({x_i},{x_j})\;d{x_1} \cdots d{x_n}} } .$$

Answering a question raised by Conlon and Lee, we prove that in order for t_G(∣h∣)^1/∣E(G)∣ to be a norm on ℋ, the graph G must be edge-transitive.