Year: 2026
Type: preprint
Abstract: Let $T_n$ be the infinite $n$-regular tree, $n\ge3$. We prove that every finite connected vertex set $Ω\subset T_n$ satisfies the sharp inequality \[ σ_1(Ω)\le \frac{n}{(n-1)|Ω|+1}. \] Equality holds if and only if $Ω$ is a ball. Since \[ |δΩ|=(n-2)|Ω|+2, \] the result is equivalently a sharp upper bound at fixed external boundary cardinality, and hence a discrete Weinstock inequality on $T_n$.
Source: arXiv (Cornell University)
Authors: Wang, Lili, Wang, Tao
Institution: Peking University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G3912457081, G7647179894
OpenAlex ID: W7212077045