Weinstock Inequality on Regular Trees

RESUMO

Let $T_n$ be the infinite $n$-regular tree, $n\ge3$. We prove that every finite connected vertex set $\Omega\subset T_n$ satisfies the sharp inequality \[ \sigma_1(\Omega)\le \frac{n}{(n-1)|\Omega|+1}. \] Equality holds if and only if $\Omega$ is a ball. Since \[ |\delta\Omega|=(n-2)|\Omega|+2, \] the result is equivalently a sharp upper bound at fixed external boundary cardinality, and hence a discrete Weinstock inequality on $T_n$.