Categorifying the magnitude of a graph
By Richard Hepworth, Simon Willerton. 2015
77 citations recorded by the source corpus.
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.
Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/9fb330600122f95c62530c1718b2d76d5c372bcf
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Computational Theory and Mathematics
In Discrete Mathematics and Combinatorics
In Geometry and Topology
- Michael Shulman
- Tom Leinster
- Mark W. Meckes
- Simon Willerton
- Richard Hepworth
- Dominic R. Verity
- Emily Riehl
- R. Haugseng
Citations in the corpus
- Magnitude homology of enriched categories and metric spaces (2017, methodology): “In [HW15], Hepworth and Willerton constructed such a homology theory for the special case of graphs, regarded as metric spaces with the shortest path metric (or equivalently as categories enriched over the sub-monoidal-category N ⊆ [0,∞]).”
- Maximizing diversity in biology and beyond (2015, unknown): “For instance, it automatically produces a notion of the Euler characteristic of an (ordinary) category, closely related to the topological Euler characteristic [14]; in the context of metric spaces, magnitude encodes geometric information such as volume and dimension [15–17]; in graph theory, magnit”
- The magnitude of a graph (2014, unknown): “I am especially grateful to David Speyer and Simon Willerton for sharing their thoughts, and for allowing me to include Proposition 5.1.”