Spectral preorder and perturbations of discrete weighted graphs

$m$-扇形离散拉普拉斯算子与复加权图的重现性

$m$-sectorial discrete Laplacians and recurrence of complex-weighted graphs

研究复加权图的Dirichlet拉普拉斯算子的m-扇形性质，将其扩展为电网络，并定义复加权图的重现性，通过收敛结果给出其多种刻画。

带权单纯复形上的Hodge拉普拉斯算子：形式、闭包与有界实现

Geometric Criteria for Essential Self-Adjointness of Discrete Hodge Laplacians on Weighted Simplicial Complexes

该研究针对带权图与单纯复形建立离散Hodge拉普拉斯算子的范数界，推导了本质自伴性，通过线复形归约得到正则图的尖锐界，并结合周期格的Floquet-Bloch计算确定了具体范数值。

加权有向图的几何与谱分析

Geometric and spectral analysis on weighted digraphs

本文用几何方法研究加权有向图的入/出拉普拉斯算子与邻接矩阵，证明零特征值重数等于源点或汇点数目，并将环流解释为离散散度的核，同时给出割容量和流值的计算公式。

离散情形下的定量 Obata 定理

Quantitative Obata's theorem in discrete setting

本文在离散图上建立了定量 Obata 定理：当加权图的 Bakry-Émery Ricci 曲率下界为正且第 d 个非零 Laplace 特征值接近该下界时，证明图在组合结构、Frobenius 距离和特征函数上均接近 d 维超立方体图。

Discrete quantitative nodal theorem

We prove a theorem that can be thought of as a common generalization of the Discrete Nodal Theorem and (one direction of) Cheeger's Inequality for graphs. special case of this result will assert that if the second and third eigenvalues of the Laplacian are at least c apart, then the subgraphs induced by the positive and negative supports of the eigenvector belonging to the second eigenvalue are not only connected, but edge-expanders (in a weighted sense, with expansion depending on c).

Counting the Number of Minimal Paths in Weighted Coloured-Edge Graphs

A weighted coloured-edge graph is a graph for which each edge is assigned both a positive weight and a discrete colour, and can be used to model transportation and computer networks in which there are multiple transportation modes. In such a graph paths are compared by their total weight in each colour, resulting in a Pareto set of minimal paths from one vertex to another. This paper will give a tight upper bound on the cardinality of a minimal set of paths for any weighted coloured-edge graph. Additionally, a bound is presented on the expected number of minimal paths in weighted bicoloured-ed…

Some structural and enumerative aspects of discrete surfaces and PL-manifolds

This manuscript recounts some of the author's contributions to algebraic and enumerative combinatorics. We have focused on two types of generalizations of bipartite maps, which are bipartite graphs embedded on surfaces. Maps are known to appear in many areas of theoretical physics and discrete mathematics, but one key interest for fundamental computer science is how multi-facetted they are in the sense that multiple encodings exist which are not interchangeable, like Tutte's/loop equations, the topological recursion, the KP hierarchy and numerous bijections which made the field so rich. One ge…