• Corpus ID: 15244476

Variants on the minimum rank problem: A survey II

@article{Fallat2011VariantsOT,
  title={Variants on the minimum rank problem: A survey II},
  author={Shaun M. Fallat and Leslie Hogben},
  journal={arXiv: Combinatorics},
  year={2011}
}
The minimum rank problem for a (simple) graph $G$ is to determine the smallest possible rank over all real symmetric matrices whose $ij$th entry (for $i\neq j$) is nonzero whenever $\{i,j\}$ is an edge in $G$ and is zero otherwise. This paper surveys the many developments on the (standard) minimum rank problem and its variants since the survey paper \cite{FH}. In particular, positive semidefinite minimum rank, zero forcing parameters, and minimum rank problems for patterns are discussed. 

Figures from this paper

On bounded rank positive semidefinite matrix completions of extreme partial correlation matrices
TLDR
An upper bound foregd(G) is shown in terms of a new tree-width-like parameter $\sla(G), defined as the smallest $r$ for which $G$ is a minor of the strong product of a tree and $K_r$ is shown.
Complexity of the positive semidefinite matrix completion problem with a rank constraint
We consider the decision problem asking whether a partial rational symmetric matrix with an all-ones diagonal can be completed to a full positive semidefinite matrix of rank at most k. We show that
A new graph parameter related to bounded rank positive semidefinite matrix completions
The Gram dimension $$\mathrm{gd}(G)$$ of a graph $$G$$ is the smallest integer $$k\ge 1$$ such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the
Orthonormal Representations of H-Free Graphs
TLDR
It is shown that for certain bipartite graphs $H$, there is a connection between the Turan number of $H$ and the maximum of $\vartheta \left( \overline{G} \right)$ over all $H-free graphs $G$.
Minimum Vector Rank and Complement Critical Graphs
Given a graph G, a real orthogonal representation of G is a function from its set of vertices to R^d such that two vertices are mapped to orthogonal vectors if and only if they are not neighbors. The
...
...

References

SHOWING 1-10 OF 99 REFERENCES
Program for calculating bounds on the minimum rank of a graph using Sage
TLDR
A program is developed using the open-source mathematics software Sage to implement several techniques of reduction that can be programmed on a computer.
Zero forcing sets and minimum rank of graphs
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
Techniques for determining the minimum rank of a small graph
Lower Bounds in Minimum Rank Problems
On the Minimum Rank Among Positive Semidefinite Matrices with a Given Graph
TLDR
Upper and lower bounds for the minimum rank of all matrices in the set of all positive semidefinite matrices whose graph is G are given and used to determineoperatorname(G) for some well-known graphs.
The minimum rank of symmetric matrices described by a graph: A survey☆
An upper bound for the minimum rank of a graph
UNIVERSALLY OPTIMAL MATRICES AND FIELD INDEPENDENCE OF THE MINIMUM RANK OF A GRAPH
The minimum rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose (i, j)th entry (for ij) isnonzero whenever {i, j} isan edge in G and is
...
...