The sharper bounds conjecture for the Neumann surplus on quantum graphs

Let $\Gamma$ be a compact metric graph, let $\partial\Gamma$ denote its boundary, let $\beta$ be its first Betti number, and let $\omega_n$ denote the Neumann surplus associated with the $n$-th eigenfunction. Sharper bounds conjecture. The Neumann surplus is bounded by

$$-1-\left|\partial\Gamma\right|\le\omega_n\le\beta+1.$$

These bounds are proposed because the previously obtained bounds appear not to be strict in many examples. The conjecture also concerns the extent to which graph topology controls Neumann surplus statistics, alongside the observation that the Neumann count can distinguish some tree graphs with equal nodal counts.