Categorifying the magnitude of a graph

Richard Hepworth, Simon Willerton

Research output: Contribution to journalArticle

1 Citation (Scopus)
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Abstract

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
Volume19
Issue number2
Early online date2 Aug 2017
DOIs
Publication statusPublished - 2017

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Graph in graph theory
Homology
Khovanov Homology
Jones Polynomial
Euler Characteristic
Theorem
Power series
Join
Metric space
Integer

Keywords

  • magnitude
  • graph
  • categorification

Cite this

Categorifying the magnitude of a graph. / Hepworth, Richard; Willerton, Simon.

In: Homology, Homotopy and Applications, Vol. 19, No. 2, 2017, p. 31-60.

Research output: Contribution to journalArticle

Hepworth, Richard ; Willerton, Simon. / Categorifying the magnitude of a graph. In: Homology, Homotopy and Applications. 2017 ; Vol. 19, No. 2. pp. 31-60.
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