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21
Parallel BlockDiagonalBordered Sparse Linear Solvers for Electrical Power System Applications
, 1995
"... This thesis presents research into parallel linear solvers for blockdiagonalbordered sparse matrices. The blockdiagonalbordered form identifies parallelism that can be exploited for both direct and iterative linear solvers. We have developed efficient parallel blockdiagonalbordered sparse dire ..."
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Cited by 11 (3 self)
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This thesis presents research into parallel linear solvers for blockdiagonalbordered sparse matrices. The blockdiagonalbordered form identifies parallelism that can be exploited for both direct and iterative linear solvers. We have developed efficient parallel blockdiagonalbordered sparse direct methods based on both LU factorization and Choleski factorization algorithms, and we have also developed a parallel blockdiagonalbordered sparse iterative method based on the GaussSeidel method. Parallel factorization algorithms for blockdiagonalbordered form matrices require a specialized ordering step coupled to an explicit load balancing step in order to generate this matrix form and to distribute the computational workload uniformly for an irregular matrix throughout a distributedmemory multiprocessor. Matrix orderings are performed using a diakoptic technique based on nodetearingnodal analysis. Parallel GaussSeidel algorithms for blockdiagonalbordered form matrices require a twopart matrix ordering technique  first to partition the matrix into blockdiagonalbordered form, again, using the nodetearing diakoptic techniques and then to multicolor the data in the last diagonal block using graph coloring techniques. The ordered matrices have extensive parallelism, while maintaining the strict precedence relationships in the GaussSeidel algorithm. Empirical
CRITICAL GROUPS AND LINE GRAPHS
"... This paper is an overview of what the author has learned about the critical group of a graph, including some new results. In particular we discuss the critical group of a graph in relation to that of its line graph when the original graph is regular. We begin by introducing the critical group from v ..."
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Cited by 5 (1 self)
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This paper is an overview of what the author has learned about the critical group of a graph, including some new results. In particular we discuss the critical group of a graph in relation to that of its line graph when the original graph is regular. We begin by introducing the critical group from various aspects. We then study the
On vertexmagic and edgemagic total injections of graphs, Australas
 J. Combin
"... The study of graph labellings has focused on finding classes of graphs which admit a particular type of labelling. Here we consider variations of the wellknown edgemagic and vertexmagic total labellings for which all graphs admit such a labelling. In particular, we consider two types of injection ..."
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Cited by 5 (1 self)
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The study of graph labellings has focused on finding classes of graphs which admit a particular type of labelling. Here we consider variations of the wellknown edgemagic and vertexmagic total labellings for which all graphs admit such a labelling. In particular, we consider two types of injections of the vertices and edges of a graph with positive integers: (1) for every edge the sum of its label and those of its endvertices is some magic constant (edgemagic); and (2) for every vertex the sum of its label and those of the edges incident to it is some magic constant (vertexmagic). Our aim is to minimise the maximum label or the magic constant associated with the injection. We present upper bounds on these parameters for complete graphs, forests and arbitrary graphs, which in a number of cases are within a constant factor of being optimal. Our results are based on greedy algorithms for computing an antimagic injection, which is then extended to a magic total injection. Of independent interest is our result that every forest has an edgeantimagic vertex labelling. 1
Magic Carpets
, 2000
"... : A settheoretic structure, the magic carpet, is defined and some of its combinatorial properties explored. The magic carpet is a generalization and abstraction of labeled diagrams such as magic squares and magic graphs, in which certain configurations of points on the diagram add to the same value ..."
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Cited by 3 (0 self)
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: A settheoretic structure, the magic carpet, is defined and some of its combinatorial properties explored. The magic carpet is a generalization and abstraction of labeled diagrams such as magic squares and magic graphs, in which certain configurations of points on the diagram add to the same value. Some basic definitions and theorems are presented as well as computergenerated enumerations of small nonisomorphic magic carpets of various kinds. Introduction In its most general form, a magic carpet is a collection of k different subsets of a set S of positive integers, where the integers in each subset sum to the same magic constant m. In this paper we always take S = {1, 2, 3, ... n}, and refer to a magic carpet on this set as an (n, k)carpet. A (9,8)carpet is shown in Figure 1, with each element of S depicted as a point (labeled with the element it represents) and each subset of S as a line connecting the points in that subset. Figure 1. A (9,8) magic carpet This is just an o...
On magic graphs
 Australasian J. Combin
"... A(p, q)graph G =(V,E) is said to be magic if there exists a bijection f: V ∪ E →{1, 2, 3,...,p+ q} such that for all edges uv of G, f(u)+ f(v) +f(uv) is a constant. The minimum of all constants say, m(G), where the minimum is taken over all such bijections of a magic graph G, is called the magic st ..."
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A(p, q)graph G =(V,E) is said to be magic if there exists a bijection f: V ∪ E →{1, 2, 3,...,p+ q} such that for all edges uv of G, f(u)+ f(v) +f(uv) is a constant. The minimum of all constants say, m(G), where the minimum is taken over all such bijections of a magic graph G, is called the magic strength of G. In this paper we define the maximum of all constants say, M(G), analogous to m(G), and introduce strong magic, ideal magic, weak magic labelings, and prove that some known classes of graphs admit such labelings. 1
The Complete Catalog of 3Regular, Diameter3 Planar Graphs
, 1996
"... The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 ..."
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The largest known 3regular planar graph with diameter 3 has 12 vertices. We consider the problem of determining whether there is a larger graph with these properties. We find all nonisomorphic 3regular, diameter3 planar graphs, thus solving the problem completely. There are none with more than 12 vertices. An Upper Bound A graph with maximum degree \Delta and diameter D is called a (\Delta; D)graph. It is easily seen ([9], p. 171) that the order of a (\Delta,D)graph is bounded above by the Moore bound, which is given by 1+ \Delta + \Delta (\Delta \Gamma 1) + \Delta \Delta \Delta + \Delta(\Delta \Gamma 1) D\Gamma1 = 8 ? ! ? : \Delta(\Delta \Gamma 1) D \Gamma 2 \Delta \Gamma 2 if \Delta 6= 2; 2D + 1 if \Delta = 2: Figure 1: The regular (3,3)graph on 20 vertices (it is unique up to isomorphism) . For D 2 and \Delta 3, this bound is attained only if D = 2 and \Delta = 3; 7, and (perhaps) 57 [3, 14, 23]. Now, except for the case of C 4 (the cycle on four vertices), the num...
Antimagic labelings of regular bipartite graphs: An application of the Marriage Theorem
, 2007
"... A labeling of a graph is a bijection from E(G) to the set {1,2,..., E(G)}. A labeling is antimagic if for any distinct vertices u and v, the sum of the labels on edges incident to u is different from the sum of the labels on edges incident to v. We say a graph is antimagic if it has an antimagic l ..."
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A labeling of a graph is a bijection from E(G) to the set {1,2,..., E(G)}. A labeling is antimagic if for any distinct vertices u and v, the sum of the labels on edges incident to u is different from the sum of the labels on edges incident to v. We say a graph is antimagic if it has an antimagic labeling. In 1990, Ringel conjectured that every connected graph other than K2 is antimagic. In this paper, we show that every regular bipartite graph (with degree at least 2) is antimagic. Our technique relies heavily on the Marriage Theorem. 1
COLORING AND LABELING PROBLEMS ON GRAPHS BY
"... This thesis studies both several extremal problems about coloring of graphs and a labeling problem on graphs. We consider colorings of graphs that are either embeddable in the plane or have low maximum degree. We consider three problems: coloring the vertices of a graph so that no adjacent vertices ..."
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This thesis studies both several extremal problems about coloring of graphs and a labeling problem on graphs. We consider colorings of graphs that are either embeddable in the plane or have low maximum degree. We consider three problems: coloring the vertices of a graph so that no adjacent vertices receive the same color, coloring the edges of a graph so that no adjacent edges receive the same color, and coloring the edges of a graph so that neither adjacent edges nor edges at distance one receive the same color. We use the model where colors on vertices must be chosen from assigned lists and consider the minimum size of lists needed to guarantee the existence of a proper coloring. More precisely, a list assignment function L assigns to each vertex a list of colors. A proper Lcoloring is a proper coloring such that each vertex receives a color from its list. A graph is klistcolorable if it has an Lcoloring for every list assignment L that assigns each vertex a list of size k. The list chromatic number χl(G) of a graph G is the minimum k such that G is klistcolorable. We also call the list chromatic number the choice number of the graph. If a graph is klistcolorable, we call it kchoosable. The elements of a graph are its vertices and edges. A proper total coloring of a graph is a coloring
Appendix: Glossary
"... F8.928> k ; : : : ; v n ]; v j \Gamma [v i ; v l ; : : : ; v m ]; : : : ; v n \Gamma [v i ; v p ; : : : v q ]]; where E = f(v i ; v j ); (v i ; v k ); : : : ; (v i ; v n ); (v j ; v l ); : : : ; (v j ; v m ); : : : ; (v n ; v p ); : : : ; (v n ; v q )g: Comment: Note that in this version any node ..."
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F8.928> k ; : : : ; v n ]; v j \Gamma [v i ; v l ; : : : ; v m ]; : : : ; v n \Gamma [v i ; v p ; : : : v q ]]; where E = f(v i ; v j ); (v i ; v k ); : : : ; (v i ; v n ); (v j ; v l ); : : : ; (v j ; v m ); : : : ; (v n ; v p ); : : : ; (v n ; v q )g: Comment: Note that in this version any node is present at least twice: as the key to each sublist<F