Pure Simplicial and Clique Complexes with a Fixed Number of Facets

TL;DR

This paper investigates the structural properties and enumeration of pure simplicial and clique complexes, providing conditions for clique complex characterization, uniqueness results, and bounds on their counts.

Contribution

It establishes a facet-based criterion for clique complexes, proves uniqueness of certain pure clique complexes from adjacency matrices, and counts complexes with fixed facets.

Findings

Facet-based characterization of clique complexes

Uniqueness of triangle-intersection free pure clique complexes from adjacency matrices

Upper bounds on the number of pure simplicial and clique complexes

Abstract

We study structural and enumerative aspects of pure simplicial complexes and clique complexes. We prove a necessary and sufficient condition for any simplicial complex to be a clique complex that depends only on the list of facets. We also prove a theorem that a class of ``triangle-intersection free" pure clique complexes are uniquely determined up to isomorphism merely from the facet-adjacency matrix. Lastly, we count the number of pure simplicial complexes with a fixed number of facets and find an upper bound to the number of pure clique complexes.