Related papers: The minimum rank problem for circulants

The minimum rank of a graph G is the minimum rank over all real symmetric matrices whose off-diagonal sparsity pattern is the same as that of the adjacency matrix of G. In this note we present the first exact algorithm for the minimum rank…

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…

For a simple graph, the minimum rank problem is to determine the smallest rank among the symmetric matrices whose off-diagonal nonzero entries occur in positions corresponding to the edges of the graph. Bounds on this minimum rank (and on…

The minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all symmetric real matrices whose $ij$th entry (for $i\neq j$) is nonzero whenever $\{i,j\}$ is an edge in $G$ and is zero otherwise. Minimum rank is a…

In this paper we introduce a new parameter for a graph called the {\it minimum universal rank}. This parameter is similar to the minimum rank of a graph. For a graph $G$ the minimum universal rank of $G$ is the minimum rank over all…

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 minimum vector rank of a graph is the…

The minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all symmetric real matrices whose $ij$th entry (for $i\neq j$) is nonzero whenever $\{i,j\}$ is an edge in $G$ and is zero otherwise. Minimum rank is a…

The minrank of a graph $G$ is the minimum rank of a matrix $M$ that can be obtained from the adjacency matrix of $G$ by switching some ones to zeros (i.e., deleting edges) and then setting all diagonal entries to one. This quantity is…

We study the minimum rank of a (simple, undirected) graph, which is the minimum rank among all matrices in a space determined by the graph. We determine the exact set of graphs on eight vertices for which the nullity of a minimum rank…

The minrank of a graph $G$ on the set of vertices $[n]$ over a field $\mathbb{F}$ is the minimum possible rank of a matrix $M\in\mathbb{F}^{n\times n}$ with nonzero diagonal entries such that $M_{i,j}=0$ whenever $i$ and $j$ are distinct…

The minimum rank of a simple graph $G$ is the smallest possible rank over all symmetric real matrices $A$ whose nonzero off-diagonal entries correspond to the edges of $G$. Using the zero forcing number, we prove that the minimum rank of…

The zero forcing number Z(G), which is the minimum number of vertices in a zero forcing set of a graph G, is used to study the maximum nullity / minimum rank of the family of symmetric matrices described by G. It is shown that for a…

The minimum skew rank $mr^{-}(\mathbb{F},G)$ of a graph $G$ over a field $\mathbb{F}$ is the smallest possible rank among all skew symmetric matrices over $\mathbb{F}$, whose ($i$,$j$)-entry (for $i\neq j$) is nonzero whenever $ij$ is an…

The minimum skew rank of a simple graph G over the field of real numbers, is the smallest possible rank among all real skew-symmetric matrices whose (i,j)-entry (for i not equal to j) is nonzero whenever {i, j} is an edge in G and is zero…

The minimum rank of a graph is the minimum possible rank of a real matrix whose zero-nonzero pattern is described by the graph. The current algorithms can compute efficiently the minimum rank of undirected trees. This paper provides an…

We study the problem of low-rank matrix completion for symmetric matrices. The minimum rank of a completion of a generic partially specified symmetric matrix depends only on the location of the specified entries, and not their values, if…

A vertex ranking of a graph is an assignment of ranks (or colors) to the vertices of the graph, in such a way that any simple path connecting two vertices of equal rank, must contain a vertex of a higher rank. In this paper we study a…

Let $G$ be an undirected graph on $n$ vertices and let $S(G)$ be the set of all $n \times n$ real symmetric matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of $G$. The inverse eigenvalue…

We provide a counterexample to a recent conjecture that the minimum rank of every sign pattern matrix can be realized by a rational matrix. We use one of the equivalences of the conjecture and some results from projective geometry. As a…

The real minimum skew rank of a simple graph G is the smallest possible rank among all real skew symmetric matrices, whose (i,j)-entry (for i not equal to j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. In this paper we…