8 problems

Let $\Gamma$ be a metrized graph, with total length $\ell(\Gamma)$ and tau constant $\tau(\Gamma)$. The ratio $\tau(\Gamma)/\ell(\Gamma)$ is invariant under scaling all edge length…

Let $\Gamma$ be a metrized graph, let $\tau(\Gamma)$ denote its tau constant, and let $\ell(\Gamma)$ denote its total length. Cinkir's explicit tau lower bound conjecture. For ever…

Let $\Gamma$ be a metrized graph, let $\ell(\Gamma)=\int_{\Gamma} dx$ be its total length, and consider metrized graphs with $\ell(\Gamma)\neq 0$. Baker and Rumely's tau lower boun…

Let $\Gamma$ be an irreducible polarized metrized graph of genus $\bar{g}\geq2$, with total length $\ell(\Gamma)$, invariants $a(\Gamma)$ and $\epsilon(\Gamma)$, and let…

Let $f:X\to Y$ be a semistable fibration, let $CV(f)$ be the set of singular fibers, and let $\Gamma_y$ be the polarized metrized graph of the fiber over $y\in CV(f)$. For each nod…

Let $f:X\to Y$ be as above, let $CV(f)$ denote the set of singular fibers, and for $y\in CV(f)$ let $\Gamma_y$ be the polarized metrized graph of the fiber. Write…

Strict $1/108$ lower bound conjecture. For every metrized graph $\Gamma$,

Baker–Rumely's lower bound conjecture. There is a universal constant $C>0$ such that, for every metrized graph $\Gamma$,