Federico Vigolo
In this note we give a short proof that graphs having no linearly small Følner sets can be partitioned into a union of expanders. We use this fact to prove a partition result for graphs admitting linearly small maximal Følner sets and we deduce that a family of such graphs must contain a family of expanders. We also show that the existence of partitions into expanders is a quasi-isometry invariant.
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Expander graphs are widely used in communication problems and construction of error correcting codes. In such graphs, information gets through very quickly. Typically, it is not true for social or biological networks, though we may find a partition of the vertices such that the induced subgraphs on them and the bipartite subgraphs between any pair of them exhibit regular behavior of information flow within or between the subsets. Implications between spectral and regularity properties are discussed.
A unified framework for the Expander Mixing Lemma for irregular graphs and its applications
本研究提出基于邻接特征值的不规则图Expander Mixing Lemma统一框架及两个新版本，采用Perron特征向量分量作顶点权重正则化图，并将其应用于推导零强迫数等NP难参数的特征值界。
Sampling from the antiferromagnetic Ising model on bipartite, regular expander graphs
本文研究了在双部正则扩张图上反铁磁伊辛模型的采样问题，发现Glauber动力学混合缓慢，但提出高效算法和配分函数近似方法。
Efficient algorithms for the Potts model on small-set expanders
本文针对小集扩张图上的铁磁 Potts 模型，提出一种近似算法，通过高效寻找伪基态并围绕其探索状态空间，解决低温问题在典型实例上的计算可行性。
Rigidity expander graphs
本文提出了图G的d维代数连通度a_d(G)的新下界，给出了图刚性的新充分条件，并证明了k正则d刚性扩展图无限族的存在性及完全图的代数连通度下界。
In this paper, we consider a relation between $k$-way expansion constant of a finite graph and the expansion constants of subgraphs in a $k$-partition of the graph. Using this relation, we show that a sequence of finite graphs which have uniformly bounded $k+1$-way expansion constants and uniformly bounded degrees can be divided into $k$ or less sequences of expanders. Furthermore, we prove that such sequence of finite graphs is not coarsely embeddable into any Hilbert space.
Algorithms for the ferromagnetic Potts model on expanders
本文提出了在最大度数为d的图上近似铁磁Potts模型配分函数的算法，主要贡献是针对低温下满足弱膨胀条件的d-正则图的完全多项式时间近似方案，并在Gibbs唯一性区域获得高效算法。
In this paper, we study the large-scale structure of dense regular graphs. This involves the notion of robust expansion, a recent concept which has already been used successfully to settle several longstanding problems. Roughly speaking, a graph is robustly expanding if it still expands after the deletion of a small fraction of its vertices and edges. Our main result allows us to harness the useful consequences of robust expansion even if the graph itself is not a robust expander. It states that every dense regular graph can be partitioned into `robust components', each of which is a robust ex…
We consider the problem of generating uniformly random partitions of the vertex set of a graph such that every piece induces a connected subgraph. For the case where we want to have partitions with linearly many pieces of bounded size, we obtain approximate sampling algorithms based on Glauber dynamics which are fixed-parameter tractable with respect to the bandwidth of $G$, with simple-exponential dependence on the bandwidth. For example, for rectangles of constant or logarithmic width this gives polynomial-time sampling algorithms. More generally, this gives sub-exponential algorithms for bo…
Modularity and Graph Expansion
研究建立图模块度与扩张性之间的等价关系，证明模块度严格小于1当且仅当图含大型扩张子图，并刻画连通分量在最优划分中被拆分的扩张阈值条件，解释模块度的分辨率极限。
Remarks on $d$-ary partitions and an application to elementary symmetric partitions
本文证明了d进制划分的新的计数公式，并展示了在基本对称划分中应用的唯一性定理。
A remark on certain restricted plane partitions and crystal melting model
本文利用Okounkov、Reshetikhin和Vafa提出的晶体熔化模型，给出了两类平面分拆配分函数的计算公式，并推导出极限边界下平面分拆配分函数的乘积公式，证明了该配分函数与双P^1模型的开弦振幅等价，同时得到了z轴方向极限边界下对称平面分拆配分函数的乘积公式。
By implementing algorithmic versions of Sapozhenko's graph container methods, we give new algorithms for approximating the number of independent sets in bipartite graphs. Our first algorithm applies to $d$-regular, bipartite graphs satisfying a weak expansion condition: when $d$ is constant, and the graph is a bipartite $Ω( \log^2 d/d)$-expander, we obtain an FPTAS for the number of independent sets. Previously such a result for $d>5$ was known only for graphs satisfying the much stronger expansion conditions of random bipartite graphs. The algorithm also applies to weighted independent sets: …
Spectral properties of random graphs with fixed equitable partition
本文定义$S$-正则图及其矩阵，推导大型随机$S$-正则矩阵的极限谱密度和特征向量统计量，并证明扩张器混合引理、Alon-Bopana界等特征值不等式。
Remarks on the countable case of the Unfriendly Partition Problem
本文研究不友好划分问题在可数图上的情形，证明当图的射线不同时经过无限多个有限度和无限度顶点时存在二分划分，并推广了先前结果。
The method of topological vertex for topological string theory on toric Calabi-Yau 3-folds is reviewed. Implications of an explicit formula of partition functions in the "on-strip" case, typically the generalized conifolds, are considered. Generating functions of part of the partition functions are shown to be tau functions of the KP hierarchy. The associated Baker-Akhiezer functions play the role of wave functions, and satisfy $q$-difference equations. These $q$-difference equations represent the quantum mirror curves conjectured by Gukov and Sułkowski.
Some remarks on polarized partition relations
本文引入广义强序列乘积的概念，通过纯组合证明其存在性可推导出极化分拆关系，拓展了Efimov提出的强序列工具的应用场景。
Remarks on a certain restricted partition function of Lin
本文通过q级数和分割公式，给出了林关于b(3n+1)和b(3n+2)生成函数的初等证明，并建立了b(n)对3的无限多个内部同余式。
Hypergraph universality via branching random walks
研究有界度均匀超图的通用性问题，沿用Alon–Capalbo图策略，结合扩张器定制积、有界密度超图分解及分支随机游走尾界，构造边数接近计数下界的通用超图。
Asymptotically Enumerating Independent Sets in Regular $k$-Partite $k$-Uniform Hypergraphs
本文将聚合物模型与团簇展开截断方法应用于满足自然扩展性质的正则 $k$-部 $k$-一致超图，渐近确定了其独立集数量，并给出了仅依赖局部结构的高效计算公式及线性超图的闭式表达式。
Remarks on MacMahon's $q$-series
本文研究MacMahon定义的分拆生成函数序列，证明非负整数k对应的A_k(q)等序列是3着色分拆函数的生成函数，C_k(q)等序列是超分拆函数的生成函数，为两类重要生成函数提供无穷多组公式。
Edge partitions into induced-$2K_2$-free bipartite graphs
本文研究了将双图边划分到诱导-2K₂-自由双图的问题，确定了最小部分数并探讨了相关界限和复杂性。
Remarks on the conjectures of Capparelli, Meurman, Primc and Primc
本文通过Griffin等人工作，将CMPP猜想与非标准特殊化标准模相关联，并探讨了CMPP分区的多个方面，包括生成函数的函数方程、层次-秩对称性和罗格斯-拉马努金型恒等式。
On partition and almost disjoint properties of combinatorial notions
本文研究半群中组合大性概念的划分与几乎不相交性质，将中心集定理的多项式扩展推广至交换可消半群，并证明 Cp-集及另三类集合的相关性质。
Absence of (complex) zeros property is at the heart of the interpolation method developed by Barvinok \cite{barvinok2017combinatorics} for designing deterministic approximation algorithms for various graph counting and computing partition functions problems. Earlier methods for solving the same problem include the one based on the correlation decay property. Remarkably, the classes of graphs for which the two methods apply sometimes coincide or nearly coincide. In this paper we show that this is more than just a coincidence. We establish that if the interpolation method is valid for a family o…
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