Symmetric nonnegative trifactorization of pattern matrices

Abstract

A factorization of an $n\times n$ nonnegative symmetric matrix A of the form $BC{B}^T$, where C is a $k\times k$ symmetric matrix, and both B and C are required to be nonnegative, is called the Symmetric Nonnegative Matrix Trifactorization (SN-Trifactorization). The SNT-rank of A is the minimal k for which such factorization exists. The SNT-rank of a simple graph G that allows loops is defined to be the minimal possible SNT-rank of all symmetric nonnegative matrices whose zero-nonzero pattern is prescribed by the graph G.

We define set-join covers of graphs, and show that finding the SNT-rank of G is equivalent to finding the minimal order of a set-join cover of G. Using this insight we develop basic properties of the SNT-rank for graphs and compute it for trees and cycles without loops. We show the equivalence between the SNT-rank for complete graphs and the Katona problem, and discuss uniqueness of patterns of matrices in the factorization.