Critical ideals, minimum rank and zero forcing number

Abstract

Recently, there have been found new relations between the zero forcing number and the minimum rank of a graph with the algebraic co-rank. We continue on this direction by giving a characterization of the graphs with real algebraic co-rank at most 2. This implies that for any graph with at most minimum rank at most 3, its minimum rank is bounded from above by its real algebraic co-rank.

Distance ideals of graphs

“…. These ideals were defined in [10] and further studied in [1,6,2,4,5], from which our study was originally inspired.…”

On graphs with 2 trivial distance ideals

“…But there are only two possibilities: either V (C 3 ) is equal to v 0 , v 1 , u 3 , u 2 , u 1 or v 0 , v 1 , v 2 , u 2 , u 1 . However, in both cases the vertices v 0 , v 1 , u 3 , u 2 , u 1 , u 4 and v −1 , v 0 , v 1 , v 2 , u 2 , u 1 would induce a 5-pan. Which is not possible since otherwise G would have three trivial distance ideals.…”

“…These were called critical ideals, see [14]. They have been explored in [2,6,7,8], and in [1,4] it was found new connections in contexts different from the Smith group or Sandpile group like the zero-forcing number and the minimum rank of a graph. In this setting, the set of forbidden graphs for the family with at most k trivial critical ideals is conjectured to be finite, see [6,Conjecture 5.5].…”

Eigenvalues, Smith normal form and determinantal ideals

“…where I n denotes the identity matrix of size n × n. The determinantal ideals associated to the matrices A X and D X have been studied in [2,3,4,5,6,17] under the name of critical ideals and distance ideals, respectively. On the other hand, the univariate determinantal ideals of A x are known as characteristic ideals.…”