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# Homology, Homotopy and Applications

## Volume 19 (2017)

### Number 2

### Categorifying the magnitude of a graph

Pages: 31 – 60

DOI: http://dx.doi.org/10.4310/HHA.2017.v19.n2.a3

#### Authors

#### 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 Künneth 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.

#### Keywords

magnitude, graph, categorification

#### 2010 Mathematics Subject Classification

05C31, 55N35

Received 15 July 2016

Published 2 August 2017