Supremal p-negative type of vertex transitive graphs

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 ⊂ ℓ 1 ( n ) ( n ⩾ 1 ) that have strict 1-negative type. The result can be stated in two ways: A subset S = { x 0 , x 1 , … , x k } of the Hamming cube { 0 , 1 } n ⊂ ℓ 1 ( n ) has generalized roundness one if and only if the vectors { x 1 − x 0 , x 2 − x 0 , … , x k − x 0 } are linearly dependent in R n . Equivalently, S has strict 1-negative type if and only if the vectors { x 1 − x 0 , x 2 − x 0 , … , x k − x 0 } are linearly independent in R n .