Homology, Homotopy and Applications

Volume 23 (2021)

Number 1

Geometric approach to graph magnitude homology

Pages: 297 – 310

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n1.a16


Yasuhiko Asao (RIKEN Center for Advanced Intelligence Project, Kyoto, Japan)

Kengo Izumihara (Graduate School of Mathematics, Kyushu University, Fukuoka, Japan)


In this paper, we introduce a new method to compute the magnitude homology of general graphs. To each direct sum component of the magnitude chain complexes, we assign a pair of simplicial complexes whose simplicial chain complex is isomorphic to it. First we state our main theorem specialized to trees, which gives another proof for the known fact that trees are diagonal. After that, we consider general graphs, which may have cycles. We also demonstrate some computation as an application.


magnitude homology, simplicial homology

2010 Mathematics Subject Classification


Received 18 March 2020

Received revised 1 July 2020

Accepted 7 July 2020

Published 4 November 2020