The maximum Laplacian ratio of a tree: a complete answer to a question of Brualdi and Goldwasser, kernel-checked in Lean
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…
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