Completely positive matrices of order five

A real matrixA of ordern is called doubly nonnegative (denotingA∈DP _n ) if it is non-negative entrywise and positive semidefinite as well.A is called completely positive (denotingA∈CP _n ) if there existk nonnegative column vectorsb ₁,b ₂,…,b _k ∈R ⁿ for some nonnegative integerk such thatA=b ₁ b′ ₁+…+b _k b′ _k . The smallest such numberk is called the factorization index ofA and is denoted by ϕ(A). This paper gives an effective criterion for any doubly nonnegative matrixA of order 5 whose associated graph is isomorphic neither toK ₅ (the complete graph) nor toK ₅−e (a subgraph ofK ₅ obtained by cutting off an edge from it) to be completely positive. This research is supported by the fund of Anhui Education Committee.