# Categorifying the magnitude of a graph

Richard Hepworth, Simon Willerton

Research output: Contribution to journalArticle

1 Citation (Scopus)

### 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 language English 31-60 30 Homology, Homotopy and Applications 19 2 2 Aug 2017 https://doi.org/10.4310/HHA.2017.v19.n2.a3 Published - 2017

### Fingerprint

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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