The maximum-degree phase-transition conjecture for optimal resistor networks

Consider optimal graphs with a prescribed average degree and $n$ vertices. Write $\Omega(n)$ for a quantity bounded below by a positive constant times $n$, and $o(n)$ for a quantity whose ratio to $n$ tends to zero.

Maximum-degree phase-transition conjecture. There is a threshold $\alpha_1$ on the average degree such that below it an optimal graph has a vertex of degree $\Omega(n)$, whereas above it all vertices have degree $o(n)$.

This is a second proposed phase transition between star-like and regular-like behaviour, addressing concentration of edges at high-degree vertices rather than merely the presence of leaves. Its status remains open in the source.