On the topology of graph picture spaces

Topology of Clique Complexes of Line Graphs

The clique complex of a graph G is a simplicial complex whose simplices are all the cliques of G, and the line graph L(G) of G is a graph whose vertices are the edges of G and the edges of L(G) are incident edges of G. In this article, we determine the homotopy type of the clique complexes of line graphs for several classes of graphs including triangle-free graphs, chordal graphs, complete multipartite graphs, wheel-free graphs, and 4-regular circulant graphs. We also give a closed form formula for the homotopy type of these complexes in several cases.

The asymptotic topology of the multineighbor complex of a random graph

We introduce the multineighbor complex of a graph, which is a simplicial complex in which a simplex is a subset of the graph with a sufficient number of mutual neighbors. We investigate the asymptotic homological properties of such complexes for the Erdos-Renyi random graphs and obtain a number of vanishing and nonvanishing results. We use this construction to perform a topological data analysis classification of noisy synthetic point clouds obtaining favorable accuracy as obtained by the standard methods. The case when there is a single neighbor has been studied earlier by Mathew Kahle.

Computational topology of graphs on surfaces

Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn on surfaces. Typical questions include representing surfaces and graphs embedded on them computationally, deciding whether a graph embeds on a surface, solving computational problems related to homotopy, optimizing curves and graphs on surfaces, and solving standard graph algorithm problems more efficiently in the case of surface-embedded graphs.

图的切割复形的拓扑

Topology of Cut Complexes of Graphs

本文定义了图的k-切割复形，推广了Alexander对偶，并利用代数拓扑、离散Morse理论等技术研究了多种图族的可壳性、同伦型和同调性质。

星拓扑下图裸表示上函数的连续性

Continuity of Functions on Bare Representation of Graphs under Star Topology

本文在有限图上引入星拓扑，将顶点和边统一为点，研究图的裸表示之间的连续映射，并指出图同态必诱导连续映射但反之不成立。

The topology of competitively constructed graphs

We consider a simple game, the $k$-regular graph game, in which players take turns adding edges to an initially empty graph subject to the constraint that the degrees of vertices cannot exceed $k$. We show a sharp topological threshold for this game: for the case $k=3$ a player can ensure the resulting graph is planar, while for the case $k=4$, a player can force the appearance of arbitrarily large clique minors.

Regularity partitions and the topology of graphons

We highlight a topological aspect of the graph limit theory. Graphons are limit objects for convergent sequences of dense graphs. We introduce the representation of a graphon on a unique metric space and we relate the dimension of this metric space to the size of regularity partitions. We prove that if a graphon has an excluded induced sub-bigraph then the underlying metric space is compact and has finite packing dimension. It implies in particular that such graphons have regularity partitions of polynomial size.

Regularity via topology of the lcm-lattice for $C_4$-free graphs

We study the topology of the lcm-lattice of edge ideals and derive upper bounds on the Castelnuovo-Mumford regularity of the ideals. In this context it is natural to restrict to the family of graphs with no induced 4-cycle in their complement. Using the above method we obtain sharp upper bounds on the regularity when the complement is a chordal graph, or a cycle, or when the primal graph is claw free with no induced 4-cycle in its complement. For the later family we show that the second power of the edge ideal has a linear resolution.

Topology of Cayley Graphs Applied to Inverse Additive Problems

We present proofs of the basic isopermetric structure theory, obtaining some new simplified proofs. As an application, we obtain simple descriptions for subsets $S$ of an abelian group with $|kS|\le k|S|-k+1$ or $|kS-rS|- (k+r)|S|,$ where $1\le r \le k.$ These results may be applied to several questions in Combinatorics and Additive Combinatorics (Frobenius Problem, Waring's problem in finite fields and Cayley graphs with a big diameter, ....).

T=0 Partition Functions for Potts Antiferromagnets on Moebius Strips and Effects of Graph Topology

We present exact calculations of the zero-temperature partition function of the $q$-state Potts antiferromagnet (equivalently the chromatic polynomial) for Moebius strips, with width $L_y=2$ or 3, of regular lattices and homeomorphic expansions thereof. These are compared with the corresponding partition functions for strip graphs with (untwisted) periodic longitudinal boundary conditions.

随机扰动图中团因子的完整图景

The complete picture for clique factors in randomly perturbed graphs

研究随机扰动图中团因子的存在概率，确定阈值p，完成对α>1/2情况的分析，证明了极端结果的稳定性版本。

总切割复形与网格图切割复形的拓扑结构

Topology of total cut complexes and cut complexes of grid graphs

本文研究网格图的总切割复形和切割复形的拓扑结构，扩展并完善了贝耶等人的工作，加强了相关猜想并深化了对这些复形性质的理解。

Topology and chromatic number of random $ε$-distance graphs on spheres

Given $0<α\leqπ$, $ε>0$ and $n$, we define random graphs on the $d$-dimensional sphere by drawing $n$ i.i.d. uniform random points for the vertices, and edges $u {\sim} v$ whenever the geodesic distance between $u$ and $v$ is $ε$-close to $α$. This model generalizes distance graphs on spheres, and also random Borsuk graphs. Topological tools are known to give tight bounds for the chromatic number of Borsuk graphs. We now study the efficiency of one of these topological invariants, namely the connectivity of Lóvasz's neighborhood complex, to bound the chromatic number of this model of random gr…