Symmetric Nonnegative Trifactorization of Pattern Matrices

A factorization of an n×n nonnegative symmetric matrix A of the form BCBT, where C is a k×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.