Supremal p-negative type of vertex transitive graphs

Abstract

We study the supremal p-negative type of connected vertex transitive graphs. The analysis provides a way to characterize subsets of the Hamming cube {0,1}n⊂ℓ1(n) (n⩾1) that have strict 1-negative type. The result can be stated in two ways: A subset S={x0,x1,…,xk} of the Hamming cube {0,1}n⊂ℓ1(n) has generalized roundness one if and only if the vectors {x1−x0,x2−x0,…,xk−x0} are linearly dependent in Rn. Equivalently, S has strict 1-negative type if and only if the vectors {x1−x0,x2−x0,…,xk−x0} are linearly independent in Rn.