Zero Forcing Sets for Graphs PDF Download

Author: Fatemeh Alinaghipour Taklimi
Publisher:
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Category :
Languages : en
Pages :

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Author: Leslie Hogben
Publisher: American Mathematical Society
ISBN: 1470466554
Category : Mathematics
Languages : en
Pages : 302

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This book provides an introduction to the inverse eigenvalue problem for graphs (IEP-$G$) and the related area of zero forcing, propagation, and throttling. The IEP-$G$ grew from the intersection of linear algebra and combinatorics and has given rise to both a rich set of deep problems in that area as well as a breadth of “ancillary” problems in related areas. The IEP-$G$ asks a fundamental mathematical question expressed in terms of linear algebra and graph theory, but the significance of such questions goes beyond these two areas, as particular instances of the IEP-$G$ also appear as major research problems in other fields of mathematics, sciences and engineering. One approach to the IEP-$G$ is through rank minimization, a relevant problem in itself and with a large number of applications. During the past 10 years, important developments on the rank minimization problem, particularly in relation to zero forcing, have led to significant advances in the IEP-$G$. The monograph serves as an entry point and valuable resource that will stimulate future developments in this active and mathematically diverse research area.

Author:
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Languages : en
Pages : 0

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In this thesis we introduce a class of regular bipartite graphs whose biadjacency matrices are circulant matrices -- a generalization of circulant graphs which happen to be bipartite -- and we describe some properties possessed by these graphs. We describe sufficient conditions for two of these graphs to be isomorphic and prove necessary conditions in some cases. We also compute upper and lower bounds for the zero forcing number of such a graph based only on the parameters that describe its biadjacency matrix. The main results of this thesis characterize the bipartite circulant graphs that achieve equality in the lower bound and compute their minimum ranks.

Author: Jakob Loedding
Publisher:
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Category :
Languages : en
Pages : 0

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Zero-forcing is a coloring game on a graph in which an initial set of vertices is colored gray while the remaining vertices are colored white. An iterative color change rule, where some vertices have the ability to force others to change their color to gray, is then applied until no more vertices may be forced. A zero-forcing set of a graph G is defined as an initial set of gray vertices in which the remaining white vertices are forced gray after some number of iterations of the color change rule. The minimum cardinality of a zero-forcing set is defined as the zero-forcing number of G. In this thesis, we define a new graph parameter called the zero-forcing diameter, which quan- tifies the minimum intersection of two minimum zero-forcing sets of a graph with respect to its zero-forcing number. Furthermore, we present the numerical bounds of the zero-forcing diameter and find its value, with theoretical proof, for specific graph families. We then introduce an integer programming model for calculating the zero-forcing number of a graph. Finally, we build upon this model to develop our own integer program for calculating the zero-forcing diameter of a graph.

Author: TeresaW. Haynes
Publisher: Routledge
ISBN: 1351454641
Category : Mathematics
Languages : en
Pages : 519

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""Presents the latest in graph domination by leading researchers from around the world-furnishing known results, open research problems, and proof techniques. Maintains standardized terminology and notation throughout for greater accessibility. Covers recent developments in domination in graphs and digraphs, dominating functions, combinatorial problems on chessboards, and more.

Author: Peter Collier
Publisher:
ISBN:
Category :
Languages : en
Pages : 0

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Zero forcing is a graph infection process where a colour change rule is applied iteratively to a graph and an initial set of vertices, S. If S results in the entire graph becoming infected, we call this set a zero forcing set. The size of the smallest zero forcing set for a graph, G, is called the zero forcing number of G. We study subgraphs of proper interval graphs to determine how the removal of edges affects the zero forcing number of these graphs. We, then, compare the zero forcing number of twisted hypercubes to that of the same size hypercube, and determine that twisted hypercubes have smaller zero forcing number. Finally, we turn our attention to probabilistic zero forcing, a variant on zero forcing, and show that there are graphs who become forced faster when initiating the process from vertices that are outside the center of the graph.

Author: Yaqi Zhang
Publisher:
ISBN:
Category : Graphs
Languages : en
Pages : 0

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Zero forcing is a color changing process on the vertices of a graph. The zero forcing number of a graph is the minimum cardinality among the zero forcing sets, which are subsets of the vertices which have the property that as a starting set it will color the whole graph. A motivation for studying zero forcing is the study of minimum rank and maximum nullity of matrices with a given sparsity pattern. The zero forcing number provides an upper bound for the maximum nullity of a matrix whose off-diagonal non-zero entries are described by the graph. The study of zero forcing includes several questions regarding, for instance, the propagation time ($\ptp(G)$) of a graph $G$ and a maximum failed zero forcing set. In this thesis, the tight upper bound $\ptp(G)\leq \lc \frac{ V(G) -\Zp(G)}{2} \rc$ is established for the positive semidefinite propagation time of a graph $G$ in terms of its positive semidefinite zero forcing number $\Zp(G)$. To prove this bound, two methods of transforming one positive semidefinite zero forcing set into another and algorithms implementing these methods are presented. Consequences of the bound, including a tight Nordhaus-Gaddum sum upper bound on the positive semidefinite propagation time, are established. In order to study the maximum failed zero forcing set, we introduce a definition for the spark of a graph which is closely related to the notion of a fort, and we build a connection from the spark of the graph to the maximum failed zero forcing set size. In addition, we consider the full spark scenario for graphs. We also study how these quantities change when we replace a graph by its shadow graph, which is obtained from a graph $G$ by adding for each vertex $v$ of $G$ a new vertex $u$, called the {shadow} vertex of $v$, and joining $u$ to the neighbors of $v$ in $G$. As the maximum nullity of a matrix is the maximum geometric multiplicity of zero as an eigenvalue, it is also natural to study the appearing multiplicities among the eigenvalues of the matrices. In the final chapter, we establish a connection between the number of distinct eigenvalues from level symmetric trees to that of a path, and present a conjecture for the minimal total eigenvalue multiplicities for a path along with some evidence in support of this conjecture.

Author: Mason Willman
Publisher:
ISBN:
Category :
Languages : en
Pages : 0

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Zero forcing is a coloring game on a graph where gray vertices can force their white neighbors based on a color change rule. A set of initial vertices is colored gray and, by a color change rule, iteratively force neighboring white vertices until only gray vertices are left. A zero forcing set is the set of initially colored vertices that force the entire graph to become gray. The zero forcing number is the minimum cardinality of the zero forcing set. Related graph parameters, such as propagation time and throttling number, provide additional information on the zero forcing game. Despite this, an integer programming model that can calculate the zero forcing number and all related graph parameters has yet to be developed. Therefore, we present a full information model that stores all information regarding the game of zero forcing. Furthermore, we prove that our model can compute optimal solutions for the zero forcing number, minimum propagation time, maximum propagation time, and throttling number. We motivate its formulation through analysis of related integer programming models and numerical experiments showcasing its computational efficiency on all isomorphic graphs of orders 3-8.

Author: Nora Hartsfield
Publisher: Courier Corporation
ISBN: 0486315525
Category : Mathematics
Languages : en
Pages : 276

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Book Description
Stimulating and accessible, this undergraduate-level text covers basic graph theory, colorings of graphs, circuits and cycles, labeling graphs, drawings of graphs, measurements of closeness to planarity, graphs on surfaces, and applications and algorithms. 1994 edition.

Author: Zhao Zhang
Publisher: Springer
ISBN: 3319126911
Category : Computers
Languages : en
Pages : 776

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Book Description
This book constitutes the refereed proceedings of the 8th International Conference on Combinatorial Optimization and Applications, COCOA 2014, held on the island of Maui, Hawaii, USA, in December 2014. The 56 full papers included in the book were carefully reviewed and selected from 133 submissions. Topics covered include classic combinatorial optimization; geometric optimization; network optimization; optimization in graphs; applied optimization; CSoNet; and complexity, cryptography, and games.