Categorifying the magnitude of a graph
Richard Hepworth, Simon Willerton · Homology Homotopy and Applications · 2017
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 Künneth 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.