Our main result is a sharp bound for the number of vertices in a minimal forbidden subgraph for the graphs having minimum rank at most 3 over the finite field of order 2. We also list all 62 such minimal forbidden subgraphs. We conclude by exploring how some of these results over the finite field of order 2 extend to arbitrary fields and demonstrate that at least one third of the 62 are minimal forbidden subgraphs over an arbitrary field for the class of graphs having minimum rank at most 3 in that field.
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 otherwise. In this paper we give an algorithm for computing the minimum skew rank of a connected unicyclic graph, and classify all connected acyclic and connected unicyclic graphs G, for which the minimum skew rank of G is equal to the minimum skew rank of P, where P is a diametrical path of G.
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 of an arbitrary graph G. In particular, we use the notion of determinantal rank to transform the minimum rank problem into a system of polynomial equations that can be solved by computational tools from algebraic geometry and commutative algebra. We provide computational results, explore possibilities for improvement, and discuss how the algorithm can be ext…
In this paper, we define the minimum (maximum) rank, term rank and the sign nonsingular of tensors. The sufficiency and necessity for the minimum rank of a real tensor to be $1$ is given. And we show that the maximum rank of a tensor is not less than the term rank. We also prove that the minimum rank of a sign nonsingular tensor is not less than the dimension of it. And we get some characterizations of a tensor having sign left or sign right inverses.
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 consequence of the counterexample, we show that there is a graph for which the minimum rank over the reals is strictly smaller than the minimum rank over the rationals. We also make some comments on the minimum rank of sign pattern matrices over different subfields of $\mathbb R$.
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.
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 study the problem of real minimum skew rank of powers and strict powers of paths.
A \emph{sign pattern (matrix)} is a matrix whose entries are from the set $\{+, -, 0\}$. The \emph{minimum rank} (respectively, \emph{rational minimum rank}) of a sign pattern matrix $\cal A$ is the minimum of the ranks of the real (respectively, rational) matrices whose entries have signs equal to the corresponding entries of $\cal A$. A sign pattern $\cal A$ is said to be \emph{condensed} if $\cal A$ has no zero row or column and no two rows or columns are identical or negatives of each other. In this paper, a new direct connection between condensed $m \times n $ sign patterns with minimum r…
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.
The structure of all graphs having minimum rank at most k over a finite field with q elements is characterized for any possible k and q. A strong connection between this characterization and polarities of projective geometries is explained. Using this connection, a few results in the minimum rank problem are derived by applying some known results from projective geometry.
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 algorithm to compute in polynomial time the minimum rank of directed trees allowing loops.
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 difficult parameter to compute. However, there are now a number of known reduction techniques and bounds that can be programmed on a computer; we have developed a program using the open-source mathematics software Sage to implement several techniques. We have also established several additional strategies for computation of minimum rank. These techniques hav…
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 smallest dimension d for which such a representation exists. This quantity is closely related to the minimum semidefinite rank of G, which has been widely studied. Considering the minimum vector rank as an analogue of the chromatic number, this work defines critical graphs as those for which the removal of any vertex decreases the minimum vector rank; and comp…
There are profound relations between the zero forcing number and minimum rank of a graph. We study the relation of both parameters with a third one, the algebraic co-rank; that is defined as the largest $i$ such that the $i$-th critical ideal is trivial. This gives a new perspective for bounding and computing these three graph parameters.
Fractional minimum positive semidefinite rank is defined from $r$-fold faithful orthogonal representations and it is shown that the projective rank of any graph equals the fractional minimum positive semidefinite rank of its complement. An $r$-fold version of the traditional definition of minimum positive semidefinite rank of a graph using Hermitian matrices that fit the graph is also presented. This paper also introduces $r$-fold orthogonal representations of graphs and formalizes the understanding of projective rank as fractional orthogonal rank. Connections of these concepts to quantum theo…
A sign pattern matrix is a matrix whose entries are from the set $\{+,-,0\}$. If $A$ is an $m\times n$ sign pattern matrix, the qualitative class of $A$, denoted $Q(A)$, is the set of all real $m\times n$ matrices $B=[b_{i,j}]$ with $b_{i,j}$ positive (respectively, negative, zero) if $a_{i,j}$ is + (respectively, $-$, 0). The minimum rank of a sign pattern matrix $A$, denoted $\mr(A)$, is the minimum of the ranks of the real matrices in $Q(A)$. Determination of the minimum rank of a sign pattern matrix is a longstanding open problem. For the case that the sign pattern matrix has a 1-separatio…
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 connected graph of order at least two, no vertex is in every zero forcing set. The positive semidefinite zero forcing number Z_+(G) is introduced, and shown to be equal to |G|-OS(G), where OS(G) is the recently defined ordered set number that is a lower bound for minimum positive semidefinite rank. The positive semidefinite zero forcing number is applied to the com…
A {\it sign pattern matrix} is a matrix whose entries are from the set $\{+,-, 0\}$. The minimum rank of a sign pattern matrix $A$ is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of $A$. It is shown in this paper that for any $m \times n$ sign pattern $A$ with minimum rank $n-2$, rational realization of the minimum rank is possible. This is done using a new approach involving sign vectors and duality. It is shown that for each integer $n\geq 9$, there exists a nonnegative integer $m$ such that there exists an $n\times m$ sign pattern…
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 difficult parameter to compute. However, there are now a number of known reduction techniques and bounds that can be programmed on a computer; we have developed a program using the open-source mathematics software Sage to implement several techniques. In this note, we provide the source code for this program.
We show that computing the minimum rank of a sign pattern matrix is NP hard. Our proof is based on a simple but useful connection between minimum ranks of sign pattern matrices and the stretchability problem for pseudolines arrangements. In fact, our hardness result shows that it is already hard to determine if the minimum rank of a sign pattern matrix is $\leq 3$. We complement this by giving a polynomial time algorithm for determining if a given sign pattern matrix has minimum rank $\leq 2$. Our result answers one of the open problems from Linial et al. [Combinatorica, 27(4):439--463, 2007].
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 butterfly network is $\frac19\left[(3r+1)2^{r+1}-2(-1)^r\right]$ and that this is equal to the rank of its adjacency matrix.
The tensor rank of some Gabidulin codes of small dimension is investigated. In particular, we determine the tensor rank of any rank metric code equivalent to an $8$-dimensional $\mathbb{F}_q$-linear generalized Gabidulin code in $\mathbb{F}_{q}^{4\times4}$. This shows that such a code is never minimum tensor rank. In this way, we detect the first infinite family of Gabidulin codes which are not minimum tensor rank.
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 edge in $G$ and is zero otherwise. We give some new properties of the minimum skew rank of a graph, including a characterization of the graphs $G$ with cut vertices over the infinite field $\mathbb{F}$ such that $mr^{-}(\mathbb{F},G)=4$, determination of the minimum skew rank of $k$-paths over a field $\mathbb{F}$, and an extending of an existing result to sho…
We investigate the expected value of various graph parameters associated with the minimum rank of a graph, including minimum rank/maximum nullity and related Colin de Verdière-type parameters. Let $G(v,p)$ denote the usual Erdős-Rényi random graph on $v$ vertices with edge probability $p$. We obtain bounds for the expected value of the random variables ${\rm mr}(G(v,p))$, ${\rm M}(G(v,p))$, $ν(G(v,p))$ and $ξ(G(v,p))$, which yield bounds on the average values of these parameters over all labeled graphs of order $v$.
The minimum rank problem is to determine for a graph $G$ the smallest rank of a Hermitian (or real symmetric) matrix whose off-diagonal zero-nonzero pattern is that of the adjacency matrix of $G$. Here $G$ is taken to be a circulant graph, and only circulant matrices are considered. The resulting graph parameter is termed the minimum circulant rank of the graph. This value is determined for every circulant graph in which a vertex neighborhood forms a consecutive set, and in this case is shown to coincide with the usual minimum rank. Under the additional restriction to positive semidefinite mat…
Let $R$ and $S$ be two sequences of nonnegative integers in nonincreasing order and with the same sum, and let ${\cal A}(R,S)$ be the class of all $(0,1)$-matrices having row sum $R$ and column sum $S$. For a positive integer $t$, the $t$-term rank of a $(0,1)$-matrix $A$ is defined as the maximum number of $1$'s in $A$ with at most one $1$ in each column, and at most $t$ $1$'s in each row. In this paper, we address conditions for the existence of a matrix in a class ${\cal A}(R,S)$ that realizes all the minimum $t$-term ranks, for $t\geq 1$.
In this paper we deal with two aspects of the minimum rank of a simple undirected graph $G$ on $n$ vertices over a finite field $\FF_q$ with $q$ elements, which is denoted by $\mr(\FF_q,G)$. In the first part of this paper we show that the average minimum rank of simple undirected labeled graphs on $n$ vertices over $\FF_2$ is $(1-\varepsilon_n)n$, were $\lim_{n\to\infty} \varepsilon_n=0$. In the second part of this paper we assume that $G$ contains a clique $K_k$ on $k$-vertices. We show that if $q$ is not a prime then $\mr(\FF_q,G)\le n-k+1$ for $4\le k\le n-1$ and $n\ge 5$. It is known that…
The spectral radius and rank of a graph are defined to be the spectral radius and rank of its adjacency matrix, respectively. It is an important problem in spectral extremal graph theory to determine the extremal graph that has the maximum or minimum spectral radius over certain families of graphs. Monsalve and Rada [Extremal spectral radius of graphs with rank 4, Linear Algebra Appl. 609 (2021) 1-11] obtained the extremal graphs with maximum and minimum spectral radii among all graphs with order n and rank 4. In this paper, we first determine the extremal graph which attains the maximum spect…
In this paper, we first determine the minimum possible size of an Fq-linear set of rank k in PG(1, q^n). We obtain this result by relating it to the number of directions determined by a linearized polynomial whose domain is restricted to a subspace. We then use this result to find a lower bound on the number of points in an Fq- linear set of rank k in PG(2, q^n). In the case k = n, this confirms a conjecture by Sziklai in [9].
Any finite simple graph $G = (V,E)$ can be represented by a collection $\mathscr{C}$ of subsets of $V$ such that $uv\in E$ if and only if $u$ and $v$ appear together in an odd number of sets in $\mathscr{C}$. Let $c_2(G)$ denote the minimum cardinality of such a collection. This invariant is equivalent to the minimum dimension of a faithful orthogonal representation of $G$ over $\mathbb{F}_2$ and is closely connected to the minimum rank of $G$. We show that $c_2(G) = \operatorname{mr}(G,\mathbb{F}_2)$ when $\operatorname{mr}(G,\mathbb{F}_2)$ is odd, or when $G$ is a forest. Otherwise, $\operat…
Let $G$ be a simple, finite, and undirected graph with vertices each given an initial coloring of either blue or white. Zero forcing on graph $G$ is an iterative process of forcing its white vertices to become blue after a finite application of a specified color-change rule. We say that an initial set $S$ of blue vertices of $G$ is a zero forcing set for $G$ under the specified color-change rule if a finite number of iterations of zero forcing results to an updated coloring where all vertices of $G$ are blue. Otherwise, we say that $S$ is a failed zero forcing set for $G$ under the specified c…
Minimum-rank parameters of complements of threshold Kneser graphs
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
该研究确定克内泽尔图$\operatorname{KG}(n,k)$在无限域上补图的对称最小秩，给出公式$\operatorname{mr}^{\mathbb F}(I(n,k)) = n - 2k + 2$，解决了AIM研讨会关于图描述矩阵族谱的一个开放问题。
Eötvös Loránd University（厄特沃什·罗兰大学）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究网络阿尔弗雷德·雷尼数学研究所）；Hokkaido University（北海道大学）；RIKEN（理化学研究所）；Osaka University（大阪大学）；Institute of Science Tokyo（东京科学大学）
A finite dynamical system (FDS) is a system of multivariate functions over a finite alphabet, that is typically used to model a network of interacting entities. The main feature of a finite dynamical system is its interaction graph, which indicates which local functions depend on which variables; the interaction graph is a qualitative representation of the interactions amongst entities on the network. As such, a major problem is to determine the effect of the interaction graph on the dynamics of the FDS. In this paper, we are interested in three main properties of an FDS: the number of images …
The positive semidefinite (psd) rank of a polytope is the size of the smallest psd cone that admits an affine slice that projects linearly onto the polytope. The psd rank of a d-polytope is at least d+1, and when equality holds we say that the polytope is psd-minimal. In this paper we develop new tools for the study of psd-minimality and use them to give a complete classification of psd-minimal 4-polytopes. The main tools introduced are trinomial obstructions, a new algebraic obstruction for psd-minimality, and the slack ideal of a polytope, which encodes the space of realizations of a polytop…
Every graph $G=(V,E)$ is an induced subgraph of some Kneser graph of rank $k$, i.e., there is an assignment of (distinct) $k$-sets $v \mapsto A_v$ to the vertices $v\in V$ such that $A_u$ and $A_v$ are disjoint if and only if $uv\in E$. The smallest such $k$ is called the Kneser rank of $G$ and denoted by $f_{\rm Kneser}(G)$. As an application of a result of Frieze and Reed concerning the clique cover number of random graphs we show that for constant $0< p< 1$ there exist constants $c_i=c_i(p)>0$, $i=1,2$ such that with high probability \[ c_1 n/(\log n)< f_{\rm Kneser}(G) < c_2 n/(\log n). \]…
An oriented graph $G^σ$ is a digraph without loops and multiple arcs, where $G$ is called the underlying graph of $G^σ$. Let $S(G^σ)$ denote the skew-adjacency matrix of $G^σ$. The rank of the skew-adjacency matrix of $G^σ$ is called the {\it skew-rank} of $G^σ$, denoted by $sr(G^σ)$. The skew-adjacency matrix of an oriented graph is skew symmetric and the skew-rank is even. In this paper we consider the skew-rank of simple oriented graphs. Firstly we give some preliminary results about the skew-rank. Secondly we characterize the oriented graphs with skew-rank 2 and characterize the oriented g…
The \emph{genus} $\mathrm{g}(G)$ of a graph $G$ is the minimum $g$ such that $G$ has an embedding on the orientable surface $M_g$ of genus $g$. A drawing of a graph on a surface is \emph{independently even} if every pair of nonadjacent edges in the drawing crosses an even number of times. The \emph{$\mathbb{Z}_2$-genus} of a graph $G$, denoted by $\mathrm{g}_0(G)$, is the minimum $g$ such that $G$ has an independently even drawing on $M_g$. By a result of Battle, Harary, Kodama and Youngs from 1962, the graph genus is additive over 2-connected blocks. In 2013, Schaefer and Štefankovič proved t…
This paper presents strong connections between four variants of the zero forcing number and four variants of the Grundy domination number. These connections bridge the domination problem and the minimum rank problem. We show that the Grundy domination type parameters are bounded above by the minimum rank type parameters. We also give a method to calculate the $L$-Grundy domination number by the Grundy total domination number, giving some linear algebra bounds for the $L$-Grundy domination number.
The power graph of a group $G$ is the graph whose vertex set is $G$ and two distinct vertices are adjacent if one is a power of the other. In this paper, the minimum degree of power graphs of certain classes of cyclic groups, abelian $p$-groups, dihedral groups and dicyclic groups are obtained. It is ascertained that the edge-connectivity and minimum degree of power graphs are equal, and consequently the minimum disconnecting sets of power graphs of the aforementioned groups are determined. Then the equality of connectivity and minimum degree of power graphs of finite groups is investigated an…
It is well-known that the zero forcing number of a graph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterize the zero forcing number of certain circulant graphs, including some bipartite circulants, cubic circulants, and circulants which are torus products, to obtain bounds on the minimum rank and the maximum nullity. We also evaluate when the zero forcing number will give equality.
Linear constraints for a matrix polytope with no fractional vertex are investigated as intersecting research among permutation codes, rank modulations, and linear programming methods. By focusing the discussion to the block structure of matrices, new classes of such polytopes are obtained from known small polytopes. This concept, called "consolidation", is applied to find a new compact graph which is known as an approach for the graph isomorphism problem. Encoding and decoding algorithms for our new permutation codes are obtained from existing algorithms for small polytopes. The minimum distan…
A transversal of a hypergraph is a set of vertices intersecting each hyperedge. We design and analyze new exponential-time algorithms to enumerate all inclusion-minimal transversals of a hypergraph. For each fixed k>2, our algorithms for hypergraphs of rank k, where the rank is the maximum size of a hyperedge, outperform the previous best. This also implies improved upper bounds on the maximum number of minimal transversals in n-vertex hypergraphs of rank k>2. Our main algorithm is a branching algorithm whose running time is analyzed with Measure and Conquer. It enumerates all minimal transver…
A finite dynamical system is a system of multivariate functions over a finite alphabet used to model a network of interacting entities. The main feature of a finite dynamical system is its interaction graph, which indicates which local functions depend on which variables; the interaction graph is a qualitative representation of the interactions amongst entities on the network. The rank of a finite dynamical system is the cardinality of its image; the periodic rank is the number of its periodic points. In this paper, we determine the maximum rank and the maximum periodic rank of a finite dynami…
