On Path Homology of Vertex Colored (Di)Graphs
In this paper, we construct the colored-path homology theory in the category of vertex colored (di)graphs and describe its basic properties. Our construction is based on the path homology theory of digraphs that was introduced in the papers of Grigoryan, Muranov, and Shing-Tung Yau and stems from th...
Main Authors: | , |
---|---|
Format: | Article |
Language: | English |
Published: |
MDPI AG
2020-06-01
|
Series: | Symmetry |
Subjects: | |
Online Access: | https://www.mdpi.com/2073-8994/12/6/965 |
id |
doaj-84515f357ad24e168f42ccb7784f0d16 |
---|---|
record_format |
Article |
spelling |
doaj-84515f357ad24e168f42ccb7784f0d162020-11-25T03:25:13ZengMDPI AGSymmetry2073-89942020-06-011296596510.3390/sym12060965On Path Homology of Vertex Colored (Di)GraphsYuri V. Muranov0Anna Szczepkowska1Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, Słoneczna 54, 10-710 Olsztyn, PolandFaculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, Słoneczna 54, 10-710 Olsztyn, PolandIn this paper, we construct the colored-path homology theory in the category of vertex colored (di)graphs and describe its basic properties. Our construction is based on the path homology theory of digraphs that was introduced in the papers of Grigoryan, Muranov, and Shing-Tung Yau and stems from the notion of the path complex. Any graph naturally gives rise to a path complex in which for a given set of vertices, paths go along the edges of the graph. We define path complexes of vertex colored (di)graphs using the natural restrictions that are given by coloring. Thus, we obtain a new collection of colored-path homology theories. We introduce the notion of colored homotopy and prove functoriality as well as homotopy invariance of homology groups. For any colored digraph, we construct the spectral sequence of colored-path homology groups which gives the effective method of computations in the general case since any (di)graph can be equipped with various colorings. We provide a lot of examples to illustrate our results as well as methods of computations. We introduce the notion of homotopy and prove functoriality and homotopy invariance of introduced vertexed colored-path homology groups. For any colored digraph, we construct the spectral sequence of path homology groups which gives the effective method of computations in the constructed theory. We provide a lot of examples to illustrate obtained results as well as methods of computations.https://www.mdpi.com/2073-8994/12/6/965colored graphpath homologyhomology spectral sequencegraph homotopy |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Yuri V. Muranov Anna Szczepkowska |
spellingShingle |
Yuri V. Muranov Anna Szczepkowska On Path Homology of Vertex Colored (Di)Graphs Symmetry colored graph path homology homology spectral sequence graph homotopy |
author_facet |
Yuri V. Muranov Anna Szczepkowska |
author_sort |
Yuri V. Muranov |
title |
On Path Homology of Vertex Colored (Di)Graphs |
title_short |
On Path Homology of Vertex Colored (Di)Graphs |
title_full |
On Path Homology of Vertex Colored (Di)Graphs |
title_fullStr |
On Path Homology of Vertex Colored (Di)Graphs |
title_full_unstemmed |
On Path Homology of Vertex Colored (Di)Graphs |
title_sort |
on path homology of vertex colored (di)graphs |
publisher |
MDPI AG |
series |
Symmetry |
issn |
2073-8994 |
publishDate |
2020-06-01 |
description |
In this paper, we construct the colored-path homology theory in the category of vertex colored (di)graphs and describe its basic properties. Our construction is based on the path homology theory of digraphs that was introduced in the papers of Grigoryan, Muranov, and Shing-Tung Yau and stems from the notion of the path complex. Any graph naturally gives rise to a path complex in which for a given set of vertices, paths go along the edges of the graph. We define path complexes of vertex colored (di)graphs using the natural restrictions that are given by coloring. Thus, we obtain a new collection of colored-path homology theories. We introduce the notion of colored homotopy and prove functoriality as well as homotopy invariance of homology groups. For any colored digraph, we construct the spectral sequence of colored-path homology groups which gives the effective method of computations in the general case since any (di)graph can be equipped with various colorings. We provide a lot of examples to illustrate our results as well as methods of computations. We introduce the notion of homotopy and prove functoriality and homotopy invariance of introduced vertexed colored-path homology groups. For any colored digraph, we construct the spectral sequence of path homology groups which gives the effective method of computations in the constructed theory. We provide a lot of examples to illustrate obtained results as well as methods of computations. |
topic |
colored graph path homology homology spectral sequence graph homotopy |
url |
https://www.mdpi.com/2073-8994/12/6/965 |
work_keys_str_mv |
AT yurivmuranov onpathhomologyofvertexcoloreddigraphs AT annaszczepkowska onpathhomologyofvertexcoloreddigraphs |
_version_ |
1724598205301129216 |