Laplacian Immanantal polynomials and the GTS poset on Trees

二部图的拉普拉斯示性多项式与图移位操作

Laplacian Immanantal Polynomials of a Bipartite Graph and Graph Shift Operation

本文将树上的图移位操作推广为广义图移位操作，据此将已知不等式推广至二部图，并利用顶点定向证明相关定理与极值问题。

由对称函数和GTS偏序集指标化的树拉普拉斯矩阵的广义矩阵多项式

Generalized Matrix polynomials of Tree Laplacians indexed by Symmetric functions and the GTS poset

本文利用任意对称函数定义矩阵的广义矩阵函数，并将树拉普拉斯矩阵多项式系数在GTS偏序集上的不等式从Schur函数推广到单项式和被遗忘对称函数情形。

Eigenvalue monotonicity of $q$-Laplacians of trees along a poset

Let $T$ be a tree on $n$ vertices with $q$-Laplacian $L_{T}^{q}$. Let $GTS_n$ be the generalized tree shift poset on the set of unlabelled trees with $n$ vertices. We prove that for all $q \in R$, going up on $GTS_n$ has the following effect: the spectral radius and the second smallest eigenvalue of $L_{T}^{q}$ increase while the smallest eigenvalue of $L_{T}^{q}$ decreases. These generalize known results for eigenvalues of the Laplacian. As a corollary, we obtain consequences about the eigenvalues of $q,t$-Laplacians and exponential distance matrices of trees.

树$q$-拉普拉斯算子的两行immanant间不等式及广义Dyck路径中的奇数高度峰

Inequalities among two rowed immanants of the $q$-Laplacian of Trees and Odd height peaks in generalized Dyck paths

本文证明了树$q$-拉普拉斯算子归一化immanant在两行划分下的不等式序列，主要引理涉及二项式系数与不可约特征标，并给出了Riordan路径及广义Dyck路径奇数高度峰的组合与概率解释。