Isometric embedding and spectral constraints for weighted graph metrics

A weighted graph $\phi G$ encodes a finite metric space $D_{\phi G}$. When is $D$ totally decomposable? When does it embed in ${\mathrm{\ell }}_1$ space? When does its representing matrix have $\le 1$ positive eigenvalue? We give useful lemmata and prove that these questions can be answered without examining $\phi$ if and only if $G$ has no $K_{2,3}$ minor. We also prove results toward the following conjecture. $D_{\phi G}$ has $\le n$ positive eigenvalues for all $\phi$, if and only if $G$ has no $K_{2,3,\mathrm{.}\mathrm{.}\mathrm{.},3}$ minor, with $n$ threes.