arXiv subjects

Leslie Hogben

Publications and source records attributed to Leslie Hogben.

At least 55 records · Page 3Linked to original sources

arXiv2017

Power domination and zero forcing

The power domination number arises from the monitoring of electrical networks and its determination is an important problem. Upper bounds for power domination numbers can be obtained by constructions. Lower bounds for the power domination number of several families of graphs are known, but they usually arise from specific properties of each family and the methods do not generalize. In this paper we exploit the relationship between power domination and zero forcing to obtain the first general lower bound for the power domination number. We apply this bound to obtain results for both the power domination of tensor products and the zero-forcing number of lexicographic products of graphs. We also establish results for the zero forcing number of tensor products and Cartesian products of graphs.

math.CO↗

arXiv2016

Power propagation time and lower bounds for power domination number

We present a counterexample to a lower bound for the power domination number given in Liao, Power domination with bounded time constraints, J. Comb. Optim. 31 (2016)725-742. We also define the power propagation time, using the power domination propagation ideas in Liao and the (zero forcing) propagation time in Hogben et al, Propagation time for zero forcing on a graph, Discrete Appl. Math.160 (2012) 1994-2005.

math.CO↗

arXiv2016

Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph

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 better understanding of all possible spectra and their associated multiplicities. These new properties are referred to as the Strong Spectral Property and the Strong Multiplicity Property. Finally, these ideas are applied to the minimum number of distinct eigenvalues associated with G, denoted by q(G). The graphs for which q(G) is at least the number of vertices of G less one are characterized.

math.CO↗

arXiv2016

Note on Nordhaus-Gaddum problems for power domination

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, our bound is tight and is also significantly better than the corresponding bound for domination number. We also improve the product upper bound for the power domination number for graphs with certain properties.

math.CO↗

arXiv2016

Note on von Neumann and Rényi entropies of a Graph

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 Rényi 2-entropy at least as great as $K_{1,n-1}$ and for $α>1$, $K_n$ maximizes Rényi $α$-entropy over graphs of order $n$. We show that adding an edge to a graph can lower its von Neumann entropy.

math.CO↗

arXiv2016

Expected values of parameters associated with the minimum rank of a graph

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 $v$.

math.CO↗

arXiv2016

Multi-part Nordhaus-Gaddum type problems for tree-width, Colin de Verdière type parameters, and Hadwiger number

A traditional Nordhaus-Gaddum problem for a graph parameter $β$ is to find a (tight) upper or lower bound on the sum or product of $β(G)$ and $β(\bar{G})$ (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 $G_1,\dots,G_r$. A traditional Nordhaus-Gaddum problem can be viewed as the special case for $r=2$ of a more general $r$-part sum or product Nordhaus-Gaddum type problem. We determine the values of the $r$-part sum and product upper bounds asymptotically as $n$ goes to infinity for the parameters tree-width and its variants largeur d'arborescence, path-width, and proper path-width. We also establish ranges for the lower bounds for these parameters, and ranges for the upper and lower bounds of the $r$-part Nordhaus-Gaddum type problems for the parameters Hadwiger number, the Colin de Verdière number $μ$ that is used to characterize planarity, and its variants $ν$ and $ξ$.

math.CO↗

arXiv2016

Rainbow arithmetic progressions

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 $aw([n],k)$ 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 of length $k$. We establish that $aw([n],3)=Θ(\log n)$ and $aw([n],k)=n^{1-o(1)}$ for $k\geq 4$. For positive integers $n$ and $k$, the expression $aw(Z_n,k)$ denotes the smallest number of colors with which elements of the cyclic group of order $n$ can be colored and still guarantee there is a rainbow arithmetic progression of length $k$. In this setting, arithmetic progressions can "wrap around," and $aw(Z_n,3)$ behaves quite differently from $aw([n],3)$, depending on the divisibility of $n$. As shown in [Jungić et al., \textit{Combin. Probab. Comput.}, 2003], $aw(Z_{2^m},3) = 3$ for any positive integer $m$. We establish that $aw(Z_n,3)$ can be computed from knowledge of $aw(Z_p,3)$ for all of the prime factors $p$ of $n$. However, for $k\geq 4$, the behavior is similar to the previous case, that is, $aw(Z_n,k)=n^{1-o(1)}$.

math.CO↗

arXiv2015

On the distance spectra of graphs

The distance matrix of a graph $G$ is the matrix containing the pairwise distances between vertices. The distance eigenvalues of $G$ are the eigenvalues of its distance matrix and they form the distance spectrum of $G$. We determine the distance spectra of halved cubes, double odd graphs, and Doob graphs, completing the determination of distance spectra of distance regular graphs having exactly one positive distance eigenvalue. We characterize strongly regular graphs having more positive than negative distance eigenvalues. We give examples of graphs with few distinct distance eigenvalues but lacking regularity properties. We also determine the determinant and inertia of the distance matrices of lollipop and barbell graphs.

math.CO↗

arXiv2015

Fractional Zero Forcing via Three-color Forcing Games

An $r$-fold analogue of the positive semidefinite zero forcing process that is carried out on the $r$-blowup of a graph is introduced and used to define the fractional positive semidefinite forcing number. Properties of the graph blowup when colored with a fractional positive semidefinite forcing set are examined and used to define a three-color forcing game that directly computes the fractional positive semidefinite forcing number of a graph. We develop a fractional parameter based on the standard zero forcing process and it is shown that this parameter is exactly the skew zero forcing number with a three-color approach. This approach and an algorithm are used to characterize graphs whose skew zero forcing number equals zero.

math.CO↗

arXiv2015

Orthogonal Representations, Projective Rank, and Fractional Minimum Positive Semidefinite Rank: Connections and New Directions

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 theory, including Tsirelson's problem, are discussed.

math.CO↗

arXiv2014

Propagation time for zero forcing on a graph

Zero forcing (also called graph infection) on a simple, undirected graph $G$ is based on the color-change rule: If each vertex of $G$ is colored either white or black, and vertex $v$ is a black vertex with only one white neighbor $w$, then change the color of $w$ to black. A minimum zero forcing set is a set of black vertices of minimum cardinality that can color the entire graph black using the color change rule. The propagation time of a zero forcing set $B$ of graph $G$ is the minimum number of steps that it takes to force all the vertices of $G$ black, starting with the vertices in $B$ black and performing independent forces simultaneously. The minimum and maximum propagation times of a graph are taken over all minimum zero forcing sets of the graph. It is shown that a connected graph of order at least two has more than one minimum zero forcing set realizing minimum propagation time. Graphs $G$ having extreme minimum propagation times $|G| - 1$, $|G| - 2$, and $0$ are characterized, and results regarding graphs having minimum propagation time $1$ are established. It is shown that the diameter is an upper bound for maximum propagation time for a tree, but in general propagation time and diameter of a graph are not comparable.

math.CO↗

arXiv2014

Variants on the minimum rank problem: A survey II

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.

math.CO↗

arXiv2014

Crossing numbers of complete tripartite and balanced complete multipartite graphs

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):= n_1/2*(n_1-1)/2*n_2/2*(n_2-1)/2. We define an analogous bound A(n_1,n_2,n_3) for the complete tripartite graph K_{n_1,n_2,n_3}, and prove that cr'(K_{n_1,n_2,n_3})\le A({n_1,n_2,n_3}). We also show that for n large enough, 0.973 A(n,n,n) \le cr'(K_{n,n,n}) and 0.666 A(n,n,n)\le cr(K_{n,n,n}), with the tighter rectilinear lower bound established through the use of flag algebras. A complete multipartite graph is balanced if the partite sets all have the same cardinality. We study asymptotic behavior of the crossing number of the balanced complete r-partite graph. Richter and Thomassen proved in 1997 that the limit as n\to\infty of cr(K_{n,n}) over the maximum number of crossings in a drawing of K_{n,n} exists and is at most 1/4. We define z(r)=3(r^2-r)/8(r^2+r-3) and show that for a fixed r and the balanced complete r-partite graph, z(r) is an upper bound to the limit superior of the crossing number divided by the maximum number of crossings in a drawing.

math.CO↗

arXiv2014

Recursive Robust PCA or Recursive Sparse Recovery in Large but Structured Noise

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 structured noise, Lt. The structure that we assume on Lt is that Lt is dense and lies in a low dimensional subspace that is either fixed or changes "slowly enough". A key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background (Lt) from moving foreground objects (St) on-the-fly. To solve the above problem, we introduce a novel solution called Recursive Projected CS (ReProCS). Under mild assumptions, we show that, with high probability (w.h.p.), ReProCS can exactly recover the support set of St at all times; and the reconstruction errors of both St and Lt are upper bounded by a time-invariant and small value at all times.

cs.IT↗

arXiv2014

Recursive Robust PCA or Recursive Sparse Recovery in Large but Structured Noise (parts 1 and 2 combined)

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 and lies in a low dimensional subspace that is either fixed or changes "slowly enough". A key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background ($L_t$) from moving foreground objects ($S_t$) on-the-fly. To solve the above problem, in recent work, we introduced a novel solution called Recursive Projected CS (ReProCS). In this work we develop a simple modification of the original ReProCS idea and analyze it. This modification assumes knowledge of a subspace change model on the $L_t$'s. Under mild assumptions and a denseness assumption on the unestimated part of the subspace of $L_t$ at various times, we show that, with high probability (w.h.p.), the proposed approach can exactly recover the support set of $S_t$ at all times; and the reconstruction errors of both $S_t$ and $L_t$ are upper bounded by a time-invariant and small value. In simulation experiments, we observe that the last assumption holds as long as there is some support change of $S_t$ every few frames.

cs.IT↗

arXiv2011

Logic circuits from zero forcing

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" as a property of each function. Such a phenomenon occurs in a circuit when the input of gates which have been already used at a given time step is further modified by a computation actually performed at a later stage. Finally, we point out that zero forcing can be also used to implement reversible computation. The model introduced here provides a potentially new tool in the analysis of Boolean functions, with particular attention to monotonicity.

cs.DM↗

arXiv2011

Zero forcing, linear and quantum controllability for systems evolving on networks

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) controllability condition. We investigate how the graph theoretic concept of a zero forcing set impacts the controllability property. In particular, we prove that if a set of vertices is a zero forcing set, the associated dynamical system is controllable. The results open up the possibility of further exploiting the analogy between networks, linear control systems theory, and quantum systems Lie algebraic theory. This study is motivated by several quantum systems currently under study, including continuous quantum walks modeling transport phenomena. Additionally, it proposes zero forcing as a new notion in the analysis of complex networks.

quant-ph↗