Homological spectral graph theory and weighted cycle counting
School of Informatics, Xiamen University（厦门大学信息学院）；Department of Mathematics and Statistics, University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校数学与统计系）
Ye Luo, Arindam Roy
备注：
Substantially revised and expanded version, with a new title; 74 pages, 6 figures. The results are extended to positive directed weights, and the period character is introduced to complete the spectral-radius extremality classification for twisted edge adjacency
Twisting a finite graph by its homology characters produces families of generally non- Hermitian weighted vertex and edge adjacency matrices. We show that these families are organized by the canonical 2-torsion character and, when it exists, a period character. The canonical character θ translates every complex character to an anti-isospectral partner, for every positive directed weight. On the unitary character torus, the untwisted matrices maximize spectral radius strictly apart from topologically forced exceptions; for edge adjacency, the only possible additional maximizer is the period character η. For symmetric weights, these results locate the two outer spectral edges of maximal-abelian and periodic Bloch families. Fourier inversion of twisted edge traces then yields weighted circuit and prime-cycle formulas in integral and finite-quotient homology classes, with θ and η, when present, governing the resulting parity effects. Thus spectral antisymmetry, spectral-edge extremality, and weighted cycle counting arise from the same homological structure.
Counting weighted independent sets beyond the permanent
Jerrum，Sinclair and Vigoda（2004）证明了任何方阵的积和式都可以在多项式时间内估计。这种计算可以被看作是近似的二分图中的边加权匹配的分区函数。等价地，这可以被视为近似二分图的线图中的顶点加权独立集的划分函数。二部图的线图是完美图，并且被称为精确的（爪，钻石，奇洞）-自由图类。那么，Jerrum、Sinclair和Vigoda的结果能延伸多远呢？我们首先表明，它扩展到（爪，奇洞）-免费的图，然后表明，它扩展到更大的类（叉，奇洞）-免费的图。我们的技术是基于图分解，这一直是最近的工作在结构图论的焦点，并在Chvatal和Sbihi（1988），Maffray和里德（1999）和Lozin和Milanic（2008）的结构结果。
Solving Combinatorial Counting Problems with Weighted First-Order Model Counting
School of Artificial Intelligence, Jilin University, Changchun, China；State Key Laboratory of Complex \& Critical Software Environment, Beihang University, China；National Research Center for Educational Materials, China；Tengen Intelligence Institute, China；Czech Technical University in Prague, Prague, Czech Republic
Counting Weighted Bi-Colored Plane Trees and Their Geometric Applications
School of Mathematical Sciences, Soochow University（苏州大学数学科学学院）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
研究整系数线性方程系统的一个中心工具是格林和陶的广义冯诺依曼定理。这个定理将计算这些方程的加权解的任务简化为计算一个特殊形式族的加权解的任务，即权$f$的Gowers范数$\Vert f \Vert_{U^{s+1}[N]}$。本文考虑实系数线性不等式组，证明了这类加权丢番图不等式的解的个数也可以由Gowers范数有界.此外，我们提供了一个充要条件的实线性型系统是由Gowers范数以这种方式。我们目前的应用程序取消的默比乌斯函数在某些序列。在本文中开发的机器可以适应的情况下，其中的权重是无界的，但适当的伪随机，与应用程序计数的解决方案的数量丢番图不等式的素数。然而，在这一背景下出现了大量额外的困难，我们就这些问题编写了一份单独的文件。
