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arXiv · 1505.04125

Categorifying the magnitude of a graph

Richard Hepworth·Simon Willerton
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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.

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BibTeXRIS

Richard Hepworth, Simon Willerton. 2015-05-15. Categorifying the magnitude of a graph. https://doi.org/10.4310/hha.2017.v19.n2.a3

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