Homology, Homotopy and Applications

Volume 20 (2018)

Number 2

On the path homology theory of digraphs and Eilenberg–Steenrod axioms

Pages: 179 – 205

DOI: https://dx.doi.org/10.4310/HHA.2018.v20.n2.a9


Alexander Grigor’yan (Department of Mathematics, University of Bielefeld, Germany; and Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia)

Rolando Jimenez (Instituto de Matematicas, National Autonomous University of Mexico (UNAM), Unidad Oaxaca, Mexico)

Yuri Muranov (Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Olsztyn, Poland)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)


In the paper we continue the investigation of the path homology theory of digraphs that was constructed in our previous papers. We prove basic theorems that are similar to the theorems of classical algebraic topology and introduce several natural constructions of digraphs which are very helpful to investigate the path homology theory. We describe relation of our results to the Eilenberg–Steenrod axiomatic of homology theory.


homology of digraphs, paths in digraphs, homotopy theory for digraphs, Eilenberg–Steenrod axioms

2010 Mathematics Subject Classification

05C20, 05C25, 05C38, 05C76, 18G60, 55N35, 55N40, 55P10, 55P40, 57M15

Received 11 September 2016

Received revised 19 January 2018

Published 30 May 2018