Homology tests for graph colorings

Knots and distributive homology: from arc colorings to Yang-Baxter homology

This paper is a sequel to my essay "Distributivity versus associativity in the homology theory of algebraic structures" Demonstratio Math., 44(4), 2011, 821-867 (arXiv:1109.4850 [math.GT]). We start from naive invariants of arc colorings and survey associative and distributive magmas and their homology with relation to knot theory. We outline potential relations to Khovanov homology and categorification, via Yang-Baxter operators. We use here the fact that Yang-Baxter equation can be thought of as a generalization of self-distributivity. We show how to define and visualize Yang-Baxter homology…

Answers to some problems about graph coloring test graphs

We prove that a graph whose chromatic number is 2 is a homotopy test graph. We also prove that there is a graph $K$ with two involutions $γ_1$ and $γ_2$ such that $(K,γ_1)$ is a Stiefel-Whitney test graph, but $(K,γ_2)$ is not. These are answers to some of the problems suggested by Kozlov.

The second homotopy group in terms of colorings of locally finite models and new results on asphericity

We describe the second homotopy group of any CW-complex $K$ by analyzing the universal cover of a locally finite model of $K$ using the notion of $G$-coloring of a partially ordered set. As applications we prove a generalization of the Hurewicz theorem, which relates the homotopy and homology of non-necessarily simply-connected complexes, and derive new results on asphericity for two-dimensional complexes and group presentations.

快速计算第一离散同调群

Fast computation of the first discrete homology group

本文提出了一种快速计算图的第一离散同调群的新算法，并通过实验验证其优于现有方法。

随机有向图的路径同调的第一Betti数

First Betti number of the path homology of random directed graphs

本研究针对随机有向图，通过组合条件刻画路径同调第一Betti数为零的稀疏与稠密区域，结合已有方法确定其为正的条件，并给出边界梯度及概率界限，同时类比应用于有向旗复形。

Can Persistent Homology provide an efficient alternative for Evaluation of Knowledge Graph Completion Methods?

In this paper we present a novel method, $\textit{Knowledge Persistence}$ ($\mathcal{KP}$), for faster evaluation of Knowledge Graph (KG) completion approaches. Current ranking-based evaluation is quadratic in the size of the KG, leading to long evaluation times and consequently a high carbon footprint. $\mathcal{KP}$ addresses this by representing the topology of the KG completion methods through the lens of topological data analysis, concretely using persistent homology. The characteristics of persistent homology allow $\mathcal{KP}$ to evaluate the quality of the KG completion looking only …