Related papers: Differential algebras on digraphs and parametrized…
In recent years, A. Grigor'yan, Y. Lin, Y. Muranov and S.T. Yau [6, 7, 8, 9] constructed a path homology theory for digraphs. Later, S. Chowdhury and F. Memoli [3] studied the persistent path homology for directed networks. In this paper,…
In this paper we introduce a primitive path homology theory on the category of simple digraphs. On the subcategory of asymmetric digraphs, this theory coincides with the path homology theory which was introduced by Grigor'yan, Lin, Muranov,…
While standard persistent homology has been successful in extracting information from metric datasets, its applicability to more general data, e.g. directed networks, is hindered by its natural insensitivity to asymmetry. We study a…
We define the notions of Reidemeister torsion and analytic torsion for directed graphs by means of the path homology theory introduced by the authors in \cite{Grigoryan-Lin-Muranov-Yau2013, Grigoryan-Lin-Muranov-Yau2014,…
We describe various path homology theories constructed for a directed hypergraph. We introduce the category of directed hypergraphs and the notion of a homotopy in this category. Also, we investigate the functoriality and the homotopy…
In this paper we study and compare two homology theories for (simple and undirected) graphs. The first, which was developed by Barcelo, Caprano, and White, is based on graph maps from hypercubes to the graph. The second theory was developed…
We introduce and study elementary properties of graph homology of algebras. This new homology theory shares many features of cyclic and Hochschild homology. We also define a graph K-theory together with an analog of Chern character.
J. Przytycki has established a connection between the Hochschild homology of an algebra $A$ and the chromatic graph homology of a polygon graph with coefficients in $A$. In general the chromatic graph homology is not defined in the case…
We develop a generalisation of the path homology theory introduced by Grigor'yan, Lin, Muranov and Yau (GLMY-theory) in a general simplicial setting. The new theory includes as particular cases the GLMY-theory for path complexes and new…
We introduce a homotopy theory of digraphs (directed graphs) and prove its basic properties, including the relations to the homology theory of digraphs constructed by the authors in previous papers. In particular, we prove the homotopy…
This paper focuses on developing an efficient algorithm for analyzing a directed network (graph) from a topological viewpoint. A prevalent technique for such topological analysis involves computation of homology groups and their…
Binary relations are one of the standard ways to encode, characterise and reason about graphs. Relation algebras provide equational axioms for a large fragment of the calculus of binary relations. Although relations are standard tools in…
For each commutative, graded algebra with finite dimension in each degree, we construct a graded cohomology theory for graphs whose graded Euler characteristic is the chromatic polynomial of the graph. This extends our previous work which…
In this article, we show that magnitude homology and path homology are closely related, and we give some applications. We define differentials ${\mathrm MH}^{\ell}_k(G) \longrightarrow {\mathrm MH}^{\ell-1}_{k-1}(G)$ between magnitude…
We prove that if two path algebras with homogeneous relations are isomorphic as algebras, then they are isomorphic as graded path algebras. This extends a result by Bell and Zhang in the connected case.
This paper studies the formal deformations of differential algebra morphisms. As a consequence, we develop a cohomology theory of differential algebra morphisms to interpret the lower degree cohomology groups as formal deformations. Then,…
A theorem of Kontsevich relates the homology of certain infinite dimensional Lie algebras to graph homology. We formulate this theorem using the language of reversible operads and mated species. All ideas are explained using a pictorial…
In this note, we interpret Leibniz algebras as differential graded Lie algebras. Namely, we consider two functors from the category of Leibniz algebras to that of differential graded Lie algebras and show that they naturally give rise to…
In this paper, we will show dichotomy theorems for the computation of polynomials corresponding to evaluation of graph homomorphisms in Valiant's model. We are given a fixed graph $H$ and want to find all graphs, from some graph class,…
Digraphs are generalizations of graphs in which each edge is assigned with a direction or two directions. In this paper, we define discrete Morse functions on digraphs, and prove that the homology of the Morse complex and the path homology…