### Abstract

Original language | English |
---|---|

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 |

### Fingerprint

### Keywords

- magnitude
- graph
- categorification

### Cite this

*Homology, Homotopy and Applications*,

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

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

Research output: Contribution to journal › Article

*Homology, Homotopy and Applications*, vol. 19, no. 2, pp. 31-60. https://doi.org/10.4310/HHA.2017.v19.n2.a3

}

TY - JOUR

T1 - Categorifying the magnitude of a graph

AU - Hepworth, Richard

AU - Willerton, Simon

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

KW - magnitude

KW - graph

KW - categorification

U2 - 10.4310/HHA.2017.v19.n2.a3

DO - 10.4310/HHA.2017.v19.n2.a3

M3 - Article

VL - 19

SP - 31

EP - 60

JO - Homology, Homotopy and Applications

JF - Homology, Homotopy and Applications

SN - 1532-0073

IS - 2

ER -