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 MayerVietoris Theorem. We prove that joins of graphs have their homology supported on the diagonal. Finally, we give various computer calculated examples.
Original language  English 

Pages (fromto)  3160 
Number of pages  30 
Journal  Homology, Homotopy and Applications 
Volume  19 
Issue number  2 
Early online date  2 Aug 2017 
DOIs  
Publication status  Published  2017 
Keywords
 magnitude
 graph
 categorification
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Richard Hepworth
 School of Natural & Computing Sciences, Mathematical Science  Senior Lecturer
 Mathematical Sciences (Research Theme)
Person: Academic