Threefold triple systems with nonsingular $N_2$

TL;DR

This paper investigates the properties of the generalized incidence matrix $N_2$ in triple systems with index three, revealing the rarity and construction methods of systems with nonsingular $N_2$ over the rationals.

Contribution

It introduces a new class of hypergraphs with nonsingular $N_2$ matrices and demonstrates their construction via PBD closure, expanding understanding of their rank properties.

Findings

Nonsingular $N_2$ systems are rare but constructible.

A range of ranks near $inom{v}{2}$ is achieved for large $v$.

Construction methods via PBD closure are provided.

Abstract

There are various results connecting ranks of incidence matrices of graphs and hypergraphs with their combinatorial structure. Here, we consider the generalized incidence matrix $N_2$ (defined by inclusion of pairs in edges) for one natural class of hypergraphs: the triple systems with index three. Such systems with nonsingular $N_2$ (over the rationals) appear to be quite rare, yet they can be constructed with PBD closure. In fact, a range of ranks near $\binom{v}{2}$ is obtained for large orders $v$.