Minimal CP rank

Abstract

Abstract. For every completely positive matrix A, cp-rank A ≥ rank A. Let cp-rank G be the maximal cp-rank of a CP matrix realization of G. Then for every graph G on n vertices, cp-rank G ≥ n. In this paper the graphs G on n vertices for which equality holds in the last inequality, and graphs G such that cp-rank A = rank A for every CP matrix realization A of G, are characterized.

Completely positive matrices over Boolean algebras and their CP-rank

“…[23] Let G be a graph on n vertices. The CP-rank of G, denoted by CP-rank(G), is the maximal CP-rank of a CP matrix realization of G, that is,…”

“…The Drew-Johnson-Loewy conjecture can be restated [23] as: for every graph G on n ≥ 4 vertices, CP-rank(G)…”

The Drew–Johnson–Loewy conjecture for matrices over max–min semirings

“…, n with {i, j} an edge if and only if i = j and a i j = 0. If A is a CP matrix and G(A) = G, we say that A is a CP matrix realization of G. Definition 1.1 [4] Let G be a graph on n vertices. The cp-rank of G, denoted by cp-rank(G), is the maximal cp-rank of a CP matrix realization of G, that is,…”

“…The Drew-Johnson-Loewy conjecture can be rephrased [4] as: For every graph G on n ≥ 4 vertices, CP-rank(G) ≤ n 2 /4 . The conjecture was proved for triangle free graphs in [3], for graphs which contain no odd cycle on five or more vertices in [5], for all graphs on five vertices which are not the complete graph in [6], for nonnegative matrices with a positive semidefinite comparison matrix (and any graph) in [7] and for all 5 × 5 CP matrices in [8].…”

A note on the CP-rank of matrices generated by Soules matrices

“…It is known that the cprank of a rank r completely positive matrix may be as big as r(r + 1)/2 − 1 (but not bigger) [1,4]. The equality cp-rank A = rank A is known to hold for completely positive matrices of rank at most 2, or of order at most 3 × 3, and for completely positive matrices with certain zero patterns [2,5].…”