Universality for Barycentric subdivision

Abstract

The spectrum of the Laplacian of successive Barycentric subdivisions of a graph converges exponentially fast to a limit which only depends on the clique number of the initial graph and not on the graph itself. The proof uses an explicit linear operator mapping the clique vector of a graph to the clique vector of the Barycentric refinement. The eigenvectors of its transpose produce integral geometric invariants for which Euler characteristic is one example.

17 pages, 2 figures

References in corpus (6)

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• Graphs with Eulerian unit spheres
• Curvature from Graph Colorings
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Cited by in corpus (16)

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• The amazing world of simplicial complexes
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• An Elementary Dyadic Riemann Hypothesis
• Dehn-Sommerville from Gauss-Bonnet
• Energized simplicial complexes
• The Cohomology for Wu Characteristics
• A Reeb sphere theorem in graph theory
• On Helmholtz free energy for finite abstract simplicial complexes
• The counting matrix of a simplicial complex
• The average simplex cardinality of a finite abstract simplicial complex
• Graph complements of circular graphs
• Division algebra valued energized simplicial complexes
• Dual graphs and modified Barlow--Bass resistance estimates for repeated barycentric subdivisions