Sum complexes - a new family of hypertrees

2009/03/07 by Nathan Linial, Linial, Nathan, Roy Meshulam +3 · 1 citation
Chemistry · Computer Science · Mathematics · #13F55 #55U10 #Combinatorics (math.CO) #Computational Drug Discovery Methods #FOS: Mathematics #Graph theory and applications #Molecular spectroscopy and chirality

paper · pdf · doi:10.48550/arxiv.0903.1359

openalex publication_date 2009/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A k-dimensional hypertree X is a k-dimensional complex on n vertices with a full (k-1)-dimensional skeleton and \binomn-1k facets such that H_k(X;Q)=0. Here we introduce the following family of simplicial complexes. Let n,k be integers with k+1 and n relatively prime, and let A be a (k+1)-element subset of the cyclic group Z_n. The sum complex X_A is the pure k-dimensional complex on the vertex set Z_n whose facets are subsets σof Z_n such that |σ|=k+1 and ∑_{x ∈ σ}x ∈ A. It is shown that if n is prime then the complex X_A is a k-hypertree for every choice of A. On the other hand, for n prime X_A is k-collapsible iff A is an arithmetic progression in Z_n.

Cited by

• An acyclic d-partition of the r-uniform complete hypergraph K_rd^(r)

Related

• Dirac's theorem on simplicial matroids
• Boundaries of Hypertrees, and Hamiltonian Cycles in Simplicial Complexes
• Pure Simplicial and Clique Complexes with a Fixed Number of Facets
• On Connectivity of the Facet Graphs of Simplicial Complexes
• A characterization of simplicial polytopes with g₂=1