Bounding the CP-rank by graph parameters

The cp-rank of a graph G, cpr(G), is the maximum cp-rank of a completely positive matrix with graph G. One obvious lower bound on cpr(G) is the (edge-) clique covering number, cc(G), i.e., the minimal number of cliques needed to cover all of G’s edges. It is shown here that for a connected graph G, cpr(G) = cc(G) if and only if G is triangle free and not a tree. Another lower bound for cpr(G) is tf(G), the maximum size of a triangle free subgraph of G. We consider the question of when does the equality cpr(G) = tf(G) hold.

Factorizations of completely positive matrices

1. Introduction. DEFINITION. If with aij = aji, is a real quadratic form inx1 …,xn, andwhere Lk = cklx1 + … + cknxn (ckj ≥ 0 for k = 1, …, t), then Q is called a completely positive form, and A = (aij) is called a completely positive matrix.

Note on Guttman Unidimensional Scaling

This paper reviews the concept of Guttman cumulative or unidimensional scaling and presents an added requirement that such scales must have a completely positive matrix of item intercorrelation. 1