Leslie Hogben

• PhD
• Iowa State University

This paper surveys results about token addition and removal (TAR) reconfiguration for several well-known vertex set parameters including domination, power domination, standard zero forcing, and PSD zero forcing. We also expand the range of parameters to which universal $X$-set TAR graph results apply, for $X$-sets and their TAR graphs introduced in...

The (disjoint) fort number and fractional zero forcing number are introduced and related to existing parameters including the (standard) zero forcing number. The fort hypergraph is introduced and hypergraph results on transversals and matchings are applied to the zero forcing number and fort number. These results are used to establish a Vizing-like...

A sign pattern is an array with entries in $\{+,-,0\}$. A real matrix $Q$ is row orthogonal if $QQ^T = I$. The Strong Inner Product Property (SIPP), introduced in [B.A. Curtis and B.L. Shader, Sign patterns of orthogonal matrices and the strong inner product property, Linear Algebra Appl. 592: 228-259, 2020], is an important tool when determining w...

In a TAR (token addition/removal) reconfiguration graph, each vertex is a set of vertices, with an edge between two vertices if one can be obtained from the other by adding or removing one element. This paper considers the X-TAR graph of a base graph G where the vertices of the X-TAR graph of G are the X-sets of G, which are subsets of V(G) that sa...

We introduce a generalization of the concept of a chronological list of forces, called a relaxed chronology. This concept is used to introduce a new way of formulating the standard zero forcing process, which we refer to as parallel increasing path covers, or PIPs. The combinatorial properties of PIPs are utilized to identify bounds comparing stand...

A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible spectra of matrices associated with $G$ is the hollow inverse eigenvalue problem for $G$. Solutions to the hollow...

The tight upper bound pt+(G)≤⌈|V(G)|−Z+(G)2⌉ is established for the positive semidefinite propagation time of a graph in terms of its positive semidefinite zero forcing number. To prove this bound, two methods of transforming one positive semidefinite zero forcing set into another and algorithms implementing these methods are presented. Consequence...

An $X$-TAR (token addition/removal) reconfiguration graph has as its vertices sets that satisfy some property $X$, with an edge between two sets if one is obtained from the other by adding or removing one element. This paper considers the $X$-TAR graph for $X-$ sets of vertices of a base graph $G$ where the $X$-sets of $G$ must satisfy certain cond...

A hollow matrix described by a graph $G$ is a real symmetric matrix having all diagonal entries equal to zero and with the off-diagonal entries governed by the adjacencies in $G$. For a given graph $G$, the determination of all possible spectra of matrices associated with $G$ is the hollow inverse eigenvalue problem for $G$. Solutions to the hollow...

The study of token addition and removal and token jumping reconfiguration graphs for power domination is initiated. Some results established here can be extended by applying the methods used for power domination to reconfiguration graphs for other parameters such as domination and zero forcing, so these results are first established in a universal...

Distance matrices of graphs were introduced by Graham and Pollack in 1971 to study a problem in communications. Since then, there has been extensive research on the distance matrices of graphs—a 2014 survey by Aouchiche and Hansen on spectra of distance matrices of graphs lists more than 150 references. In the last 10 years, variants such as the di...

Throttling addresses the question of minimizing the sum or the product of the resources used in a graph searching process and the time needed to complete the process. The study of throttling began with the study of sum throttling, and parameters that have been studied include various types of zero forcing, power domination, and cops and robbers. Re...

Distance matrices of graphs were introduced by Graham and Pollack in 1971 to study a problem in communications. Since then, there has been extensive research on the distance matrices of graphs -- a 2014 survey by Aouchiche and Hansen on spectra of distance matrices of graphs lists more than 150 references. In the last ten years, variants such as th...

The Workshop for Women in Graph Theory and Applications was held at the Institute for Mathematics and Its Applications (University of Minnesota, Minneapolis) on August 19-23, 2019. During this five-day workshop, 42 participants performed collaborative research, in six teams, each focused on open problems in different areas of graph theory and its a...

Throttling addresses the question of minimizing the sum or the product of the resources used to accomplish a task and the time needed to complete that task for various graph searching processes. Graph parameters of interest include various types of zero forcing, power domination, and Cops and Robbers. We provide a survey of product throttling for t...

A unified approach to the determination of eigenvalues and eigenvectors of specific matrices associated with directed graphs is presented. Matrices studied include the new distance matrix, with natural extensions to the distance Laplacian and distance signless Laplacian, in addition to the new adjacency matrix, with natural extensions to the Laplac...

The product power throttling number of a graph is defined to study product throttling for power domination. The domination number of a graph is an upper bound for its product power throttling number. It is established that the two parameters are equal for certain families including paths, cycles, complete graphs, unit interval graphs, and grid grap...

This paper begins the study of reconfiguration of zero forcing sets, and more specifically, the zero forcing graph. Given a base graph $G$, its zero forcing graph, $\mathscr{Z}(G)$, is the graph whose vertices are the minimum zero forcing sets of $G$ with an edge between vertices $B$ and $B'$ of $\mathscr{Z}(G)$ if and only if $B$ can be obtained f...

The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is $\mathcal{D}^L (G)=T(G)-\mathcal{D} (G)$, where $T(G)$ is the diagonal matrix of row sums of $\mathcal{D}(G)$. Several general methods are established for producing $\mathcal{D}^L$-cospectral gra...

Zero forcing is a coloring game played on a graph where each vertex is initially colored blue or white and the goal is to color all the vertices blue by repeated use of a (deterministic) color change rule starting with as few blue vertices as possible. Probabilistic zero forcing yields a discrete dynamical system governed by a Markov chain. Since i...

A unified approach to the determination of eigenvalues and eigenvectors of specific matrices associated with directed graphs is presented. Matrices studied include the distance, distance Laplacian, and distance signless Laplacian, in addition to the adjacency, Laplacian, and signless Laplacian. Various sums of Kronecker products of nonnegative matr...

Zero forcing is a process that colors the vertices of a graph blue by starting with some vertices blue and applying a color change rule. Throttling minimizes the sum of the number of initial blue vertices and the time to color the graph. In this paper, we study throttling for skew zero forcing. We characterize the graphs of order $n$ with skew thro...

Zero forcing is a coloring game played on a graph where each vertex is initially colored blue or white and the goal is to color all the vertices blue by repeated use of a (deterministic) color change rule starting with as few blue vertices as possible. Probabilistic zero forcing yields a discrete dynamical system governed by a Markov chain. Since i...

Connections between vital linkages and zero forcing are established. Specifically, the notion of a rigid linkage is introduced as a special kind of unique linkage and it is shown that spanning forcing paths of a zero forcing process form a spanning rigid linkage and thus a vital linkage. A related generalization of zero forcing that produces a rigi...

Power domination in graphs arises from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A power dominating set of a graph is a set of vertices that observes every vertex in the graph, following a set of rules for power system monitoring. A practical problem of interest is to determi...

The cop throttling number $th_c(G)$ of a graph $G$ for the game of Cops and Robbers is the minimum of $k + capt_k(G)$, where $k$ is the number of cops and $capt_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games in which both players play optimally. In this paper, we answer in the negative...

Given a real symmetric n×n matrix, the sepr-sequence t1⋯tn records information about the existence of principal minors of each order that are positive, negative, or zero. This paper extends the notion of the sepr-sequence to matrices whose entries are of prescribed signs, that is, to sign patterns. A sufficient condition is given for a sign pattern...

Zero forcing is a coloring game played on a graph that was introduced more than ten years ago in several different applications. The goal is to color all the vertices blue by repeated use of a (deterministic) color change rule. Probabilistic zero forcing was introduced by Kang and Yi in [Probabilistic zero forcing in graphs, Bull. Inst. Combin. App...

The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is $\mathcal{D}^L(G)=T(G)-\mathcal{D}(G)$, where $T(G)$ is the diagonal matrix of row sums of $\mathcal{D}(G)$. We establish several general methods for producing $\mathcal{D}^L$-cospectral graphs t...

The concept of zero forcing is extended from graphs to uniform hypergraphs in analogy with the way zero forcing was defined as an upper bound for the maximum nullity of the family of symmetric matrices whose nonzero pattern of entries is described by a given graph: A family of symmetric hypermatrices is associated with a uniform hypergraph and zero...

The conjecture of Graham and Lov ́asz that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal is proved; it is also shown that the (normalized) coefficients are log-concave. Upper and lower bounds on the location of the peak are established.

Connections between vital linkages and zero forcing are established. Specifically, the notion of a rigid linkage is introduced as a special kind of unique linkage and it is shown that spanning forcing paths of a zero forcing process form a spanning rigid linkage and thus a vital linkage. A related generalization of zero forcing that produces a rigi...

Zero forcing is a process on a graph that colors vertices blue by starting with some of the vertices blue and applying a color change rule. Throttling minimizes the sum of the size of the initial blue vertex set and the number of the time steps needed to color the graph. We study throttling for positive semidefinite zero forcing. We establish a tig...

Given a real symmetric $n\times n$ matrix, the sepr-sequence $t_1\cdots t_n$ records information about the existence of principal minors of each order that are positive, negative, or zero. This paper extends the notion of the sepr-sequence to matrices whose entries are of prescribed signs, that is, to sign patterns. A sufficient condition is given...

Zero forcing is a process on a graph that colors vertices blue by starting with some of the vertices blue and applying a color change rule. Throttling minimizes the sum of the size of the initial blue vertex set and the number of the time steps needed to color the graph. We study throttling for positive semidefinite zero forcing. We establish a tig...

The maximum nullity of a simple graph G, denoted M(G), is the largest possible nullity over all symmetric real matrices whose ijth entry is nonzero exactly when fi, jg is an edge in G for i =6 j, and the iith entry is any real number. The zero forcing number of a simple graph G, denoted Z(G), is the minimum number of blue vertices needed to force a...

We consider the cop-throttling number of a graph $G$ for the game of Cops and Robbers, which is defined to be the minimum of $(k + \text{capt}_k(G))$, where $k$ is the number of cops and $\text{capt}_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games. We provide some tools for bounding the...

We consider the cop-throttling number of a graph $G$ for the game of Cops and Robbers, which is defined to be the minimum of $(k + \text{capt}_k(G))$, where $k$ is the number of cops and $\text{capt}_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games. We provide some tools for bounding the...

Power domination in graphs arises from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A power dominating set of a graph is a set of vertices that observes every vertex in the graph, following a set of rules for power system monitoring. A practical problem of interest is to determi...

We establish new bounds on the minimum number of distinct eigenvalues among real symmetric matrices with nonzero off-diagonal pattern described by the edges of a graph and apply these to determine the minimum number of distinct eigenvalues of several families of graphs and small graphs.

The inverse eigenvalue problem of a given graph $G$ is to determine all possible spectra of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in $G$. Barrett et al. introduced the Strong Spectral Property (SSP) and the Strong Multiplicity Property (SMP) in [8]. In that paper it was shown that if a graph has a matrix...

The inverse eigenvalue problem of a given graph $G$ is to determine all possible spectra of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in $G$. Barrett et al. introduced the Strong Spectral Property (SSP) and the Strong Multiplicity Property (SMP) in [8]. In that paper it was shown that if a graph has a matrix...

The enhanced principal rank characteristic sequence (epr-sequence) of an n x n matrix is a sequence l1l2…..ln, where each lk is A, S, or N according as all, some, or none of its principal minors of order k are nonzero. There has been substantial work on epr-sequences of symmetric matrices (especially real symmetric matrices) and real skew-symmetric...

Zero forcing is an iterative coloring procedure on a graph that starts by initially coloring vertices white and blue and then repeatedly applies the following rule: if any blue vertex has a unique (out-)neighbor that is colored white, then that neighbor is forced to change color from white to blue. An initial set of blue vertices that can force the...

We conjecture that all connected graphs of order n have von Neumann entropy at least as great as the star K-1,K-n-1 and prove this for almost all graphs of order n. We show that connected graphs of order n have Renyi 2-entropy at least as great as K-1,K-n-1 and for alpha > 1, K-n maximizes Renyi alpha-entropy over graphs of order n. We show that ad...

Zero-forcing and power domination are iterative processes on graphs where an initial set of vertices are observed, and new vertices becomes observed based on simple rules. In both cases, the goal is to eventually observe the entire graph using the fewest number of initial vertices. Chang et al. introduced $k$-power domination in [Generalized power...

Zero forcing and power domination are iterative processes on graphs where an initial set of vertices are observed, and additional vertices become observed based on some rules. In both cases, the goal is to eventually observe the entire graph using the fewest number of initial vertices. Chang et al. introduced $k$-power domination in [Generalized po...

The upper and lower Nordhaus-Gaddum bounds over all graphs for the power domination number follow from known bounds on the domination number and examples. In this note we improve the upper sum bound for the power domination number substantially for graphs having the property that both the graph and its complement must be connected. For these graphs...

The upper and lower Nordhaus-Gaddum bounds over all graphs for the power domination number follow from known bounds on the domination number and examples. In this note we improve the upper sum bound for the power domination number substantially for graphs having the property that both the graph and its complement must be connected. For these graphs...

We conjecture that all connected graphs of order $n$ have von Neumann entropy at least as great as the star $K_{1,n-1}$ and prove this for almost all graphs of order $n$. We show that connected graphs of order $n$ have R\'enyi 2-entropy at least as great as $K_{1,n-1}$ and for $\alpha>1$, $K_n$ maximizes R\'enyi $\alpha$-entropy over graphs of orde...

In this paper, we investigate the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions. For positive integers $n$ and $k$, the expression $\mathrm{aw}([n],k)$ denotes the smallest number of colors with which the integers $\{1,\ldots,n\}$ can be colored and still guarantee there is a rainbow arithmetic progression...

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 Verdi\`ere-type parameters. Let $G(v,p)$ denote the usual Erd\H{o}s-R\'enyi random graph on $v$ vertices with edge probability $p$. We obtain bounds for the expected value of the rand...

A traditional Nordhaus-Gaddum problem for a graph parameter $\beta$ is to find a (tight) upper or lower bound on the sum or product of $\beta(G)$ and $\beta(\bar{G})$ (where $\bar{G}$ denotes the complement of $G$). An $r$-decomposition $G_1,\dots,G_r$ of the complete graph $K_n$ is a partition of the edges of $K_n$ among $r$ spanning subgraphs $G_...

For a given graph G and an associated class of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G, the collection of all possible spectra for such matrices is considered. Building on the pioneering work of Colin de Verdiere in connection with the Strong Arnold Property, two extensions are devised that target a b...

The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric matrix is a sequence where is A, S, or N according as all, some, or none of its principal minors of order k are nonzero. Such sequences give more information than the (0,1) pr-sequences previously studied (where basically the kth entry is 0 or 1 according as none or a...

A loop graph S is a finite undirected graph that allows loops but does not allow multiple edges. The set S(S) of real symmetric matrices associated with a loop graph of order n is the set of symmetric matrices A = [a(ij)] is an element of R-nxn such that a(ij) not equal 0 if and only if ij is an element of E(S). The minimum (maximum) rank of a loop...

The crossing number cr(G) of a graph G is the minimum number of crossings in a nondegenerate planar drawing of G. The rectilinear crossing number cr'(G) of G is the minimum number of crossings in a rectilinear nondegenerate planar drawing (with edges as straight line segments) of G. Zarankiewicz proved in 1952 that cr'(K_{n_1,n_2})\le Z(n_1,n_2):=...

Given an n×nn×n matrix, its principal rank characteristic sequence is a sequence of length n+1n+1 of 0s and 1s where, for k=0,1,…,nk=0,1,…,n, a 1 in the kth position indicates the existence of a principal submatrix of rank k and a 0 indicates the absence of such a submatrix. The principal rank characteristic sequences for symmetric matrices over va...

Associated with a simple graph G is a family of real, symmetric zero diagonal matrices with the same nonzero pattern as the adjacency matrix of G. The minimum of the ranks of the matrices in this family is denoted mr 0 (G). We characterize all connected graphs G with extreme minimum zero-diagonal rank: a connected graph G has mr 0 (G) ≤ 3 if and on...

Barrett et al. asked in W. Barrett et al. Minimum rank of edge subdivisions of graphs. Electronic Journal of Linear Algebra, 18:530-563, 2009.], whether the maximum nullity is equal to the zero forcing number for all complete subdivision graphs. We prove that this equality holds. Furthermore, we compute the value of M(F, G) = Z(G) by introducing th...

This work studies the recursive robust principal components analysis (PCA) problem. If the outlier is the signal-of-interest, this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors, $S_t$, in the presence of large but structured noise, $L_t$. The structure that we assume on $L_t$ is that $L_t$ is dense an...

A simple digraph describes the off-diagonal zero-nonzero pattern of a family of (not necessarily symmetric) matrices. Minimum rank of a simple digraph is the minimum rank of this family of matrices; maximum nullity is defined analogously. The simple digraph zero forcing number is an upper bound for maximum nullity. Cut-vertex reduction formulas for...

We establish the bounds 4/3 <= b(v) <= b(xi) <= root 2, where b(v) and b(xi) are the Nordhaus-Gaddum sum upper bound multipliers, i.e., v(G) + v((G) over bar) <= b(v)vertical bar G vertical bar and xi(G) + xi((G) over bar) <= b(xi)vertical bar G vertical bar for all graphs G, and v and xi are Colin de Verdiere type graph parameters. The Nordhaus-Ga...

The positive semidefinite zero forcing number Z + (G) of a graph G was introduced in F. Barioli et al. [ibid. 433, No. 2, 401–411 (2010; Zbl 1209.05139)]. We establish a variety of properties of Z + (G): Any vertex of G can be in a minimum positive semidefinite zero forcing set (this is not true for standard zero forcing). The graph parameters tw (...

We introduce the study of potentially eventually exponentially positive (PEEP) sign patterns and establish several results using the connections between these sign patterns and the potentially eventually positive (PEP) sign patterns. It is shown that the problem of characterizing PEEP sign patterns is not equivalent to that of characterizing PEP si...

Tree‐width, and variants that restrict the allowable tree decompositions, play an important role in the study of graph algorithms and have application to computer science. The zero forcing number is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by a graph. We establish relationships between these param...

This work studies the recursive robust principal components' analysis(PCA) problem. Here, "robust" refers to robustness to both independent and correlated sparse outliers. If the outlier is the signal-of-interest, this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors, St, in the presence of large but str...

The zero forcing number, maximum nullity and path cover number of a (simple, undirected) graph are parameters that are important in the study of minimum rank problems. We investigate the effects on these graph parameters when an edge is subdivided to obtain a so-called edge subdivision graph. An open question raised by W. Barrett et al. [Electron....

Zero forcing (also called graph infection) on a simple, undirected graph GG is based on the color-change rule: if each vertex of GG is colored either white or black, and vertex vv is a black vertex with only one white neighbor ww, then change the color of ww to black. A minimum zero forcing set is a set of black vertices of minimum cardinality that...

The minimum rank of a graph has been an interesting and well studied parameter investigated by many researchers over the past decade or so. One of the many unresolved questions on this topic is the so-called graph complement conjecture, which grew out of a workshop in 2006. This conjecture asks for an upper bound on the sum of the minimum rank of a...

A new class of sign patterns contained in the class of sign patterns that allow eventual nonnegativity is introduced and studied. A sign pattern is potentially strongly eventually nonnegative (PSEN) if there is a matrix with this sign pattern that is eventually nonnegative and has some power that is both nonnegative and irreducible. Using Perron-Fr...

The maximum positive semidefinite nullity of a multigraph G is the largest possible nullity over all real positive semidefinite matrices whose (i, j)th entry (for i&\neq$j) is zero if i and j are not adjacent in G, is nonzero if {i, j} is a single edge, and is any real number if {i, j} is a multiple edge. The definition of the positive semidefinite...

Potentially eventually positive (PEP) sign patterns were introduced by A. Berman et al. [Electron. J. Linear Algebra 19, 108–120 (2010; Zbl 1190.15031)], where it was noted that a matrix is PEP if its positive part is primitive, and an example was given of a 3×3 PEP sign pattern with reducible positive part. We extend these results by constructing...

We study the dynamics of systems on networks from a linear algebraic perspective. The control theoretic concept of controllability describes the set of states that can be reached for these systems. Under appropriate conditions, there is a connection between the quantum (Lie theoretic) property of controllability and the linear systems (Kalman) cont...

Zero forcing (also called graph infection) on a simple, undirected graph G is based on the color-change rule. The color-change rule states that if each vertex of G colored either white or black and vertex v is a black vertex with only one white neighbor u then v forces u to become black. A zero forcing set is a set of black vertices that can force...

For a simple graph G the zero forcing number Z(G) is the minimum number of black vertices initially needed to force all vertices in G black according to the color change rule. The color change rule states that for G with all vertices colored either black or white, if a vertex v is black and an adjacent white vertex w is the only white neighbor of v...

A graph is denoted G=(V,E), where V is a nonempty set of vertices, E is a set of edges, and each edge is two-element subset of the set of vertices V. Our research focuses on simple undirected graphs and a type of graph parameter called the zero forcing number. A specific type of zero forcing number, positive semidefinite zero forcing number, was in...

Zero forcing is a process of changing the color of vertices on a graph G from white to black following a color change rule. The zero forcing number, Z(G), is the minimum number of initially black vertices needed to obtain an all black final coloring. A real symmetric matrix A=[aij] is described by G if, for i≠j, aij≠0 whenever i and j are adjacent...

We design logic circuits based on the notion of zero forcing on graphs; each gate of the circuits is a gadget in which zero forcing is performed. We show that such circuits can evaluate every monotone Boolean function. By using two vertices to encode each logical bit, we obtain universal computation. We also highlight a phenomenon of "back forcing"...

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 pape...

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≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. Maximum nullity is taken over the same set of matrices, and the sum of maximum nullity and minimum rank is the order of the graph. The zero for...

Rado showed that a rational matrix is partition regular over N if and only if it satisfies the columns condition. We investigate linear algebraic properties of the columns condition, especially for oriented (vertex-arc) incidence matrices of directed graphs and for sign pattern matrices. It is established that the oriented incidence matrix of a dir...

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 gr...

A matrix A is power-positive if some positive integer power of A is entrywise positive. A sign pattern A is shown to require power-positivity if and only if either A or −A is nonnegative and has a primitive digraph, or equivalently, either A or −A requires eventual positivity. A sign pattern A is shown to be potentially power-positive if and only i...

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 = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise; maximum nullity is taken over the same set of matrices. The zero forcing number is the minimum size of a zero forcing set of vertices and bou...

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...

Several necessary or sufficient conditions for a sign pattern to allow eventual posi-tivity are established. It is also shown that certain families of sign patterns do not allow eventual positivity. These results are applied to show that for n ≥ 2, the minimum number of positive entries in an n × n sign pattern that allows eventual positivity is n...

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 deVerdì ere-type parameters. Let G(v, p) denote the usual Erd˝ os-Rényi random graph on v vertices with edge probability p. We obtain bounds for the expected value of the random variable...

The minimum rank of a sign pattern matrix is defined to be the smallest possible rank over all real matrices having the given sign pattern. Maximum nullity of a sign pattern is the largest possible nullity over the same set of matrices, and is equal to the number of columns minus the minimum rank of the sign pattern. Definitions of various graph pa...