### 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 |
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Pages (from-to) | 31-60 |

Number of pages | 30 |

Journal | Homology, Homotopy and Applications |

Volume | 19 |

Issue number | 2 |

Early online date | 2 Aug 2017 |

DOIs | |

Publication status | Published - 2017 |

### Keywords

- magnitude
- graph
- categorification

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## Profiles

### Richard Hepworth

- School of Natural & Computing Sciences, Mathematical Science - Senior Lecturer
- Mathematical Sciences (Research Theme)

Person: Academic

## Cite this

Hepworth, R., & Willerton, S. (2017). Categorifying the magnitude of a graph.

*Homology, Homotopy and Applications*,*19*(2), 31-60. https://doi.org/10.4310/HHA.2017.v19.n2.a3