Categorifying the magnitude of a graph

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.

arXiv comment29 pages, one figure and many tables. v2: minor changes

References in corpus (6)

• The magnitude of metric spaces
• A categorification for the chromatic polynomial
• On the asymptotic magnitude of subsets of Euclidean space
• On the magnitude of spheres, surfaces and other homogeneous spaces
• Magnitude, diversity, capacities, and dimensions of metric spaces
• A categorification for the Tutte polynomial

Cited by in corpus (22)

• The magnitude of a metric space: from category theory to geometric measure theory
• Magnitude meets persistence. Homology theories for filtered simplicial sets
• The magnitude of a graph
• Persistent Magnitude
• Magnitude homology of metric spaces and order complexes
• Magnitude homology of enriched categories and metric spaces
• Graph magnitude homology via algebraic Morse theory
• Magnitude homology of geodesic metric spaces with an upper curvature bound
• On the path homology of Cayley digraphs and covering digraphs
• Entropy and Diversity: The Axiomatic Approach
• On the magnitude and intrinsic volumes of a convex body in Euclidean space
• Thin Posets, CW Posets, and Categorification
• Quantales, persistence, and magnitude homology
• Alpha magnitude
• Girth, magnitude homology, and phase transition of diagonality
• On finite generation in magnitude (co)homology, and its torsion
• Smoothness filtration of the magnitude complex
• Magnitude Homology, Diagonality, Medianness, Künneth and Mayer-Vietoris
• Geometric approach to graph magnitude homology
• Magnitude homology equivalence of Euclidean sets
• Magnitude homology of geodesic space
• Magnitude homology of graphs and discrete Morse theory on Asao-Izumihara complexes