Critical ideals, minimum rank and zero forcing number

Critical ideals, minimum rank and zero forcing number

“…. Therefore, as noted in [2], it follows by the Weak Nullstellensatz that if R is an algebraically closed field, then mr R (G) ≤ γ R (G). That is not the case for the integers, there exist graphs for which mr Z (G) > γ Z (G).…”

“…Further, it is also a generalization of several other algebraic objects like Smith group or characteristic polynomials of the adjacency and Laplacian matrices, see [6,Section 4] and [13,Section 3.3]. In [2], there were explered its relation with the zero forcing number and the minimum rank. We continue on this direction.…”

“…In [7], it was proved that mz(G) ≤ mr R (G) for any field R. And in [2], it was proved that mz(G) ≤ γ R (G) for any commutative ring R with unity. However, the relation between mr R (G) and γ R ′ (G) depends on the rings R and R ′ .…”

“…That is not the case for the integers, there exist graphs for which mr Z (G) > γ Z (G). For the field of real numbers, it was conjectured [2] that mr R (G) ≤ γ R (G). Trying to sheed some light on this conjecture, it was proved in [2]…”

“…For the field of real numbers, it was conjectured [2] that mr R (G) ≤ γ R (G). Trying to sheed some light on this conjecture, it was proved in [2]…”