Discrete Concavity of Token-Graph Spectral Radii via Lorentzian Semigroups

Abstract

Let $F_k(G)$ be the $k$-token graph of a finite graph with nonnegative edge weights, and let $A_k$ and $D_k$ be its weighted adjacency and degree matrices. For every $-1\le \vartheta \le 1$, we prove that $k\mapsto {\lambda }_{\max }(A_k+\vartheta D_k)$ is discretely concave. Complement symmetry then makes this sequence nondecreasing up to the middle level. At $\vartheta =1$ and $\vartheta =0$, this gives the signless-Laplacian and adjacency spectral-radius monotonicity conjectures of Apte, Parekh, and Sud. The spectral result follows from a finite-time theorem: for every $t\ge 0$, the heat contents ${\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^{-1}{1}^{\ast }{e}^{t(A_k+\vartheta D_k)}1$ are log-concave in $k$. We encode all token levels in one Lorentzian polynomial. A four-variable operator symbol proves preservation by each edge heat gate, and the Lie-Trotter formula passes this preservation to the full semigroup. Large-time growth rates recover the top eigenvalues. The same construction also yields Lorentzian polynomials from top spectral projections. Finally, we show that the local symbol certifies exactly the parameter range $[-1,1]$.

arXiv comment11 pages; no figures