Hepworth, R. and Willerton, S. (2017) Categorifying the magnitude of a graph. Homology, Homotopy and Applications, 19 (2). pp. 31-60. ISSN 1532-0073
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.
Metadata
Item Type: | Article |
---|---|
Authors/Creators: |
|
Copyright, Publisher and Additional Information: | © 2017, Richard Hepworth and Simon Willerton. |
Keywords: | magnitude; graph; categorification |
Dates: |
|
Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 16 May 2017 10:17 |
Last Modified: | 24 Jan 2020 16:57 |
Status: | Published |
Publisher: | International Press |
Refereed: | Yes |
Identification Number: | 10.4310/HHA.2017.v19.n2.a3 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:116307 |