The Minimum Rank of Symmetric Matrices Described by a Graph: A Survey
نویسندگان
چکیده
The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i 6= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. This paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues.
منابع مشابه
Minimum Rank Problems
A graph describes the zero-nonzero pattern of a family of matrices, with the type of graph (undirected or directed, simple or allowing loops) determining what type of matrices (symmetric or not necessarily symmetric, diagonal entries free or constrained) are described by the graph. The minimum rank problem of the graph is to determine the minimum among the ranks of the matrices in this family; ...
متن کاملMinimum rank of skew - symmetric matrices described by a graph ∗ IMA - ISU research group on minimum rank
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 6= 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 amon...
متن کاملMinimum Rank of Matrices Described by a Graph or Pattern over the Rational, Real and Complex Numbers
We use a technique based on matroids to construct two nonzero patterns Z1 and Z2 such that the minimum rank of matrices described by Z1 is less over the complex numbers than over the real numbers, and the minimum rank of matrices described by Z2 is less over the real numbers than over the rational numbers. The latter example provides a counterexample to a conjecture in [AHKLR] about rational re...
متن کاملEla on the Minimum Rank of Not Necessarily Symmetric Matrices: a Preliminary Study∗
The minimum rank of a directed graph Γ is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i, j) is an arc in Γ and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i = j) is nonzero whenever {i, j} is an edge in G and is zero o...
متن کاملVariants on the minimum rank problem: A survey II∗
1 The minimum rank problem for a (simple) graph G is to determine the smallest possible 2 rank over all real symmetric matrices whose ijth entry (for i 6= j) is nonzero whenever {i, j} 3 is an edge in G and is zero otherwise. This paper surveys the many developments on the 4 (standard) minimum rank problem and its variants since the survey paper [36]. In particular, 5 positive semidefinite mini...
متن کامل