First and Second Order Signatures of Extreme Uniform Hypergraphs and Their Relationship with Vectors of the Vertex Degrees

Abstract

The adjacency matrices of extremal 3-uniform hypergraphs occupy a significant amount of computer memory. The solution of two problems is considered: to propose an efficient way of representing and storing such matrices and to find fast algorithms that allow us to operate just with vectors of the vertex degrees and signatures (characteristics of adjacency matrices), without using adjacency matrices in memory. As part of the first task, a second-order signature that uniquely defines an extremal 3-uniform hypergraph without using its adjacency matrix is described. A mechanism for compressing the second-order signature is also proposed, which contributes to greater storage efficiency. For the second problem, a number of algorithms are presented to describe the relationship between the vector of the vertex degrees and signatures of both the first and second orders. In addition, it is shown that an arbitrary second-order signature constructed under a number of constraints always has an extremal 3-uniform hypergraph corresponding to it.