Negative type gap of theta-containing metric graphs

Determine the infimum of the negative type gap \(\Gamma(M,d)\) over all theta-containing metric graphs whose edge lengths are at least \(1\).

Background

For a metric space $(M,d)$, the negative type gap $\Gamma(M,d)$ is defined as the supremum of $\gamma_d(\omega)$ over finitely supported functions $\omega$ of zero total mass and normalized $\ell_1$-norm. The paper proves that every theta-containing metric graph with all edge lengths at least $1$ has negative type gap at least $1/432$, while observing an upper bound of $1$ for any family containing a unit-length theta.

The unresolved issue is to determine the exact value of this infimum, thereby sharpening the bounds established in the paper for the quantitative failure of negative type among theta-containing metric graphs.