Families of nested graphs with compatible symmetric-group actions

Eric Ramos, Graham White

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

For fixed positive integers n and k, the Kneser graph KGn , k has vertices labeled by k-element subsets of { 1 , 2 , ⋯ , n} and edges between disjoint sets. Keeping k fixed and allowing n to grow, one obtains a family of nested graphs, each of which is acted on by a symmetric group in a way which is compatible with these inclusions and the inclusions of each symmetric group into the next. In this paper, we provide a framework for studying families of this kind using the FI-module theory of Church et al. (Duke Math J 164(9):1833–1910, 2015), and show that this theory has a variety of asymptotic consequences for such families of graphs. These consequences span a range of topics including enumeration, concerning counting occurrences of subgraphs, topology, concerning Hom-complexes and configuration spaces of the graphs, and algebra, concerning the changing behaviors in the graph spectra.

Original languageEnglish
Article number70
JournalSelecta Mathematica, New Series
Volume25
Issue number5
DOIs
StatePublished - 1 Dec 2019

Keywords

  • FI-modules
  • Graph theory
  • Kneser graphs
  • Representation stability

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