Categorifying the magnitude of a graph

TL;DR

This paper introduces a bigraded homology theory for graphs that categorifies the graph magnitude, providing new algebraic tools and properties analogous to classical topological theorems.

Contribution

It presents the first categorification of graph magnitude via a bigraded homology theory, extending properties like Kunneth and Mayer-Vietoris theorems.

Findings

Homology supported on the diagonal for joins of graphs

Properties of magnitude categorify to algebraic theorems

Computer examples illustrating the 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.