Quantum / CONCEPT

Sharp global spectral bound for root-to-leaf Laplacian transfer on two-level trees

On the homogeneous two-level rooted tree T(k, l) of order n = 1 + k(l + 1) with k >= 2 hubs and l >= 1 leaves per hub, the continuous-time Laplacian propagator from the root to any leaf decomposes across the three eigenvalues 0, a, b of the three-dimensional cell-constant invariant subspace. Its distinct-eigenvalue spectral entry bound equals 2 / (a d), where d = b - a = sqrt((k - l)^2 + 4 l). Scaled by n^(3/2), the squared spectral bound is dominated through an algebraic chord identity and rational square decompositions by the universal constant C_* = 2 + 6 sqrt(5) / 5, with equality on B_{k, l}^2 if and only if (k, l) = (2, 1).