tropical semiring, symmetric matrix, rank
In this paper, an upper bound for the CP-rank of a matrix over a tropical semiring is obtained, according to the vertex clique cover of the graph prescribed by the positions of zero entries in the matrix. The graphs that beget the matrices with the lowest possible CP-ranks are studied, and it is proved that any such graph must have its diameter equal to $2$.
Dolžan, David and Oblak, Polona.
"Bounds for the Completely Positive Rank of a Symmetric Matrix over a Tropical Semiring",
Electronic Journal of Linear Algebra,
Volume 34, pp. 152-162.