通过洛伦兹半群的令牌图谱半径的离散凹性
Discrete Concavity of Token-Graph Spectral Radii via Lorentzian Semigroups
- Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明带权图的令牌图谱半径关于层数具有离散凹性，从而解决Apte等人提出的单调性猜想，方法是将所有层编码为洛伦兹多项式并利用半群热传播。
AI中文摘要：
设$F_k(G)$为具有非负边权的有限图的$k$-令牌图，并设$A_k$和$D_k$为其加权邻接矩阵和度矩阵。对于每个$-1\leq\vartheta\leq1$，我们证明$k\mapsto\lambda_{\max}(A_k+\vartheta D_k)$是离散凹的。互补对称性随后使得该序列在中层之前非递减。在$\vartheta=1$和$\vartheta=0$时，这给出了Apte、Parekh和Sud关于无符号拉普拉斯和邻接谱半径单调性的猜想。该谱结果源于一个有限时间定理：对于每个$t\geq0$，热含量$\binom{n}{k}^{-1}\mathbf{1}^{*}e^{t(A_k+\vartheta D_k)}\mathbf{1}$在$k$上是对数凹的。我们将所有令牌级别编码在一个洛伦兹多项式中。一个四变量算子符号证明了每个边热门的保持性，而Lie-Trotter公式将该保持性传递给整个半群。长时间增长率恢复了最大特征值。相同的构造还从顶部谱投影中产生洛伦兹多项式。最后，我们表明局部符号恰好证明了参数范围$[-1,1]$。
英文摘要：
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 $-1\leq\vartheta\leq1$, we prove that $k\mapstoλ_{\max}(A_k+\vartheta D_k)$ 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 $t\geq0$, the heat contents $\binom{n}{k}^{-1}\mathbf{1}^{*}e^{t(A_k+\vartheta D_k)}\mathbf{1}$ 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 $[-1,1]$.
