Categorifying the magnitude of a graph
Combinatorics
2020-07-13 v2 Algebraic Topology
Category Theory
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.
Keywords
Cite
@article{arxiv.1505.04125,
title = {Categorifying the magnitude of a graph},
author = {Richard Hepworth and Simon Willerton},
journal= {arXiv preprint arXiv:1505.04125},
year = {2020}
}
Comments
29 pages, one figure and many tables. v2: minor changes