# Moscow Mathematical Journal

Volume 24, Issue 3, July–September 2024 pp. 373–390.

Cyclic Path Homology of (Di)Graphs

**Authors**:
Rolando Jimenez (1) and Yuri Muranov (2)

**Author institution:**(1) Instituto de Matematicas, UNAM. Unidad Oaxaca, 68000 Oaxaca, Mexico

(2) Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, Sloneczna 54 Street, 10-710 Olsztyn, Poland

**Summary: **

We construct a cyclic path homology theory on digraphs and graphs and describe properties of the introduced homology groups. As an intermediate step, we define the path homology theory on digraphs, which is based on non-self-intersecting paths. We compare the obtained theories with the standard path homology theory and provide examples of computations. Afterwards we apply the obtained results to the investigation of non-self-intersecting cycles in colored (di)graphs.

2020 Math. Subj. Class. 55N35, 05C20, 05C38, 05C25, 05C22, 55U15.

**Keywords:**Path complex, digraph, path homology, cycles in graph, cycles in digraph, colored digraphs, non-self-intersecting paths, nonself-intersecting cycles, colored graphs.

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