Symmetric Nonnegative Trifactorization of Pattern Matrices

A factorization of an $n \times n$ nonnegative symmetric matrix $A$ of the form $BCB^T$, 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 a given graph. 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.