Cheeger inequalities on simplicial complexes
Abstract
Cheegertype inequalities in which the decomposability of a graph and the spectral gap of its Laplacian mutually control each other play an important role in graph theory and network analysis, in particular in the context of expander theory. The natural problem, however, to extend such inequalities to simplicial complexes and their higher order Eckmann Laplacians has been open for a long time. The question is not only to prove an inequality, but also to identify the right Cheegertype constant in the first place. Here, we solve this problem. Our solution involves and combines constructions from simplicial topology, signed graphs, Gromov filling radii and an interpolation between the standard 2Laplacians and the analytically more difficult 1Laplacians, for which, however, the inequalities become equalities. It is then natural to develop a general theory for $p$Laplacians on simplicial complexes and investigate the related Cheegertype inequalities.
 Publication:

arXiv eprints
 Pub Date:
 February 2023
 DOI:
 10.48550/arXiv.2302.01069
 arXiv:
 arXiv:2302.01069
 Bibcode:
 2023arXiv230201069J
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Geometric Topology;
 Mathematics  Metric Geometry;
 Mathematics  Spectral Theory
 EPrint:
 Comments are welcome