Discrete Concavity of Token-Graph Spectral Radii via Lorentzian Semigroups
Abstract
Let Fk(G) be the k-token graph of a finite graph with nonnegative edge weights, and let Ak and Dk be its weighted adjacency and degree matrices. For every −1≤ϑ≤1, we prove that k↦λmax(Ak+ϑ Dk) is discretely concave. Complement symmetry then makes this sequence nondecreasing up to the middle level. At ϑ=1 and ϑ=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≥0, the heat contents binomnk−11*et(Ak+ϑ Dk)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].
Related
- Lorentzian polynomials and log-concavity of the independence polynomials of graphs
- Positive expansiveness of operator semigroups: orbit growth and spectral criteria
- Spectral characterizations of stable operator semigroups
- Generators of stability-preserving semigroups, spectral gaps, and classical ground-state computation
- The Algebraic Connectivity and Laplacian Spectral Radius of Token Graphs
BibTeX
@misc{jiang2026discrete,
title = {{Discrete Concavity of Token-Graph Spectral Radii via Lorentzian Semigroups}},
author = {Weiqi Jiang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25037},
eprint = {2609.25037},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25037},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Discrete Concavity of Token-Graph Spectral Radii via Lorentzian Semigroups
AU - Weiqi Jiang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.25037
UR - https://arxiv.org/abs/2609.25037
AB - 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≤ϑ≤1, we prove that k↦λ_max(A_k+ϑ D_k) is discretely concave. Complement symmetry then makes this sequence nondecreasing up to the middle level. At ϑ=1 and ϑ=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≥0, the heat contents binomnk^−11^*e^t(A_k+ϑ 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].
ER -