Categorifying the magnitude of a graph

Richard Hepworth, Simon Willerton

Research output: Contribution to journalArticlepeer-review

18 Citations (Scopus)
15 Downloads (Pure)


The magnitude of a graph can be thought of as an integer power series associated to a graph; Leinster introduced it using his idea of magnitude of a metric space. Here we introduce a bigraded homology theory for graphs which has the magnitude as its graded Euler characteristic. This is a categorification of the magnitude in the same spirit as Khovanov homology is a categorification of the Jones polynomial. We show how properties of magnitude proved by Leinster categorify to properties such as a Kunneth Theorem and a Mayer-Vietoris Theorem. We prove that joins of graphs have their homology supported on the diagonal. Finally, we give various computer calculated examples.
Original languageEnglish
Pages (from-to)31-60
Number of pages30
JournalHomology, Homotopy and Applications
Issue number2
Early online date2 Aug 2017
Publication statusPublished - 2017


  • magnitude
  • graph
  • categorification


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