On Path Homology of Vertex Colored (Di)Graphs

Abstract

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.