Abstract
AbstractIn this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.
Highlights
In this paper, we continue to study the homological properties of colored digraphs and graphs
The path homology theory is a homology theory for digraphs that computes the simplicial homology of a finite simplicial complex S if applied to its incidence digraph G = GS defined in the following way
The cohomology theory of digraphs that is dual to the path homology theory was introduced in previous studies [5,6,7,8]
Summary
We continue to study the homological properties of colored digraphs and graphs. The cohomology theory of digraphs that is dual to the path homology theory was introduced in previous studies [5,6,7,8]. This cohomology theory was motivated by the physical applications of discrete mathematics. This theory provides a differential calculus on digraphs and discrete sets which are considered as discretizations of topological spaces.
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