Homology, Homotopy and Applications
Volume 19 (2017)
Categorifying the magnitude of a graph
Pages: 31 – 60
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.
magnitude, graph, categorification
2010 Mathematics Subject Classification
Paper received on 15 July 2016.