# Categorifying the magnitude of a graph

@article{Hepworth2015CategorifyingTM, title={Categorifying the magnitude of a graph}, author={Richard A. Hepworth and Simon Willerton}, journal={arXiv: Combinatorics}, year={2015} }

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…

## 29 Citations

Torsion in the magnitude homology of graphs

- MathematicsJournal of Homotopy and Related Structures
- 2019

Magnitude homology is a bigraded homology theory for finite graphs defined by Hepworth and Willerton, categorifying the power series invariant known as magnitude which was introduced by Leinster. We…

The magnitude of a graph

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2017

Abstract The magnitude of a graph is one of a family of cardinality-like invariants extending across mathematics; it is a cousin to Euler characteristic and geometric measure. Among its…

A categorification of the Vandermonde determinant

- MathematicsJournal of Knot Theory and Its Ramifications
- 2022

In the spirit of Bar-Natan’s construction of Khovanov homology, we give a categorification of the Vandermonde determinant. Given a sequence of positive integers [Formula: see text], we construct a…

Graph magnitude homology via algebraic Morse theory

- Mathematics
- 2018

We compute magnitude homology of various graphs using algebraic Morse theory. Specifically, we (1) give an alternative proof that trees are diagonal, (2) identify a new class of diagonal graphs, (3)…

Magnitude homology of graphs and discrete Morse theory on Asao-Izumihara complexes

- Mathematics
- 2021

Recently, Asao and Izumihara introduced CW-complexes whose homology groups are isomorphic to direct summands of the graph magnitude homology group. In this paper, we study the homotopy type of the…

Magnitude meets persistence. Homology theories for filtered simplicial sets

- Mathematics
- 2018

The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely…

Girth, magnitude homology, and phase transition of diagonality

- Mathematics
- 2021

This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. Magnitude and magnitude homology are formulations of the Euler…

Magnitude homology of enriched categories and metric spaces

- MathematicsAlgebraic & Geometric Topology
- 2021

Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as $[0,\infty)$-enriched categories. We show that in many cases magnitude can be categorified to a…

Magnitude cohomology

- MathematicsMathematische Zeitschrift
- 2022

Magnitude homology was introduced by Hepworth and Willerton in the case of graphs, and was later extended by Leinster and Shulman to metric spaces and enriched categories. Here we introduce the dual…

Geometric approach to graph magnitude homology

- Mathematics
- 2020

In this paper, we introduce a new method to compute magnitude homology of general graphs. To each direct sum component of magnitude chain complexes, we assign a pair of simplicial complexes whose…

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