{"id":"hexagon/2610.00565","pdf":"https://hexagonmath.org/pdf/2610.00565v1","url":"https://hexagonmath.org/abs/2610.00565v1","tags":["cs.LO","msc:05C05","msc:05C35","msc:05C50","msc:15A15","msc:68V20","Laplacian permanent","Lean 4","Mathlib","extremal graph theory"],"score":0,"title":"The maximum Laplacian ratio of a tree: a complete answer to a question of Brualdi and Goldwasser, kernel-checked in Lean","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Peter W. Murphy, MD"],"reviews":0,"subject":"math.CO","summary":"For a tree $T$ with Laplacian $L(T)$, Brualdi and Goldwasser (1984) introduced the Laplacian ratio $\\pi(T)=\\operatorname{per} L(T)/\\prod_v\\deg v$ and asked which tree on $n$ vertices maximizes it. We answer the question for every $n$. For every $n\\ge 4$ the maximum is attained by a spider of cherry arms: a centre carrying pendant paths of length two (cherries) and arms, an arm being a vertex that carries cherries. For $n\\ge 492$ the centre carries only arms, almost all with five cherries, and the exceptional arms are fixed by $6(n-1)\\bmod 11$ through an explicit rule; for $4\\le n\\le 491$ the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude"],"discussed":0,"published":"2026-10-27","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}