Weinstock Inequality on Regular Trees

Let $T_n$ be the infinite $n$-regular tree, $n\ge 3$. We prove that every finite connected vertex set $\mathrm{\Omega }\subset T_n$ satisfies the sharp inequality $${\sigma }_1(\mathrm{\Omega })\le \frac{n}{(n-1)\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }+1}\mathrm{.}$$ Equality holds if and only if $\mathrm{\Omega }$ is a ball. Since $$\mathrm{\mid }\delta \mathrm{\Omega }\mathrm{\mid }=(n-2)\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }+2,$$ the result is equivalently a sharp upper bound at fixed external boundary cardinality, and hence a discrete Weinstock inequality on $T_n$.