Minimum rank of skew-symmetric matrices described by a graph

Abstract

The minimum (symmetric) rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose ijth entry (for i = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. The problem of determining minimum (symmetric) rank has been studied extensively. We define the minimum skew rank of a simple graph G to be the smallest possible rank among all skew-symmetric matrices over F whose ijth entry (for i = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. We apply techniques from the minimum (symmetric) rank problem and from skew-symmetric matrices to obtain results about the minimum skew rank problem.

The Many Face(t)s of Zero Forcing

“…Thus, zero forcing number can be used as a combinatorial bound on the minimum rank of a symmetric matrix pattern. Similar procedures have been developed for other types of matrix patterns (e.g., ones that are positivesemidefinite [EEH + 13] or skew-symmetric [IIrgomr10]) using other types of graphs and color change rules; nonsymmetric versions of zero forcing such as signed zero forcing have also been considered [GB14]. While both the graph-based parameters and the matrix-based parameters are difficult to compute, the former are generally much more tractable than the latter.…”

Grundy domination and zero forcing in Kneser graphs

“…(See the survey on minimum rank, where many applications and other interesting results on this parameter can be found [12].) In addition, the skew zero forcing number (denoted by Z − (G)) was introduced in [18], and studied in the context of the invariant mr 0 , which is a version of the minimum rank in which matrices are in addition required to have empty diagonals. Motivated by the results of [5], Lin [20] noticed a similar connection between the Grundy total domination number and the skew zero forcing number of graphs, and also between the Grundy domination number and another version of the minimum rank parameter, denoted by mr˙ .…”

On the minimum skew rank of graphs

“…The minimum skew rank problem involves skew symmetric matrices and its study began recently 1 . If the characteristic of F is 2, then a skew-symmetric matrix over F is also symmetric.…”