Graphs of order two less than the Moore bound
- Authors: Miller, Mirka , Simanjuntak, Rinovia
- Date: 2008
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 308, no. 13 (2008), p. 2810-2821
- Full Text: false
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- Description: The Moore bound for a directed graph of maximum out-degree d and diameter k is Md,k=1+d+d2++dk. It is known that digraphs of order Md,k (Moore digraphs) do not exist for d>1 and k>1. Similarly, the Moore bound for an undirected graph of maximum degree d and diameter k is . Undirected Moore graphs only exist in a small number of cases. Mixed (or partially directed) Moore graphs generalize both undirected and directed Moore graphs. In this paper, we shall show that all known mixed Moore graphs of diameter k=2 are unique and that mixed Moore graphs of diameter k3 do not exist.
- Description: C1
On the degrees of a strongly vertex-magic graph
- Authors: Balbuena, Camino , Barker, Ewan , Das, K. C. , Lin, Yuqing , Miller, Mirka , Ryan, Joe , Slamin, , Sugeng, Kiki Ariyanti , Tkac, M.
- Date: 2006
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 306, no. 6 (2006), p. 539-551
- Full Text: false
- Reviewed:
- Description: Let G=(V,E) be a finite graph, where |V|=n≥2 and |E|=e≥1. A vertex-magic total labeling is a bijection λ from V∪E to the set of consecutive integers {1,2,...,n+e} with the property that for every v∈V, λ(v)+∑w∈N(v)λ(vw)=h for some constant h. Such a labeling is strong if λ(V)={1,2,...,n}. In this paper, we prove first that the minimum degree of a strongly vertex-magic graph is at least two. Next, we show that if 2e≥10n2-6n+1, then the minimum degree of a strongly vertex-magic graph is at least three. Further, we obtain upper and lower bounds of any vertex degree in terms of n and e. As a consequence we show that a strongly vertex-magic graph is maximally edge-connected and hamiltonian if the number of edges is large enough. Finally, we prove that semi-regular bipartite graphs are not strongly vertex-magic graphs, and we provide strongly vertex-magic total labeling of certain families of circulant graphs. © 2006 Elsevier B.V. All rights reserved
- Description: C1
- Description: 2003001603
Characterization of eccentric digraphs
- Authors: Gimbert, Joan , Lopez, Nacho , Miller, Mirka , Ryan, Joe
- Date: 2006
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 306, no. 2 (2006), p. 210-219
- Full Text: false
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- Description: The eccentric digraph ED(G) of a digraph G represents the binary relation, defined on the vertex set of G, of being 'eccentric'; that is, there is an arc from u to v in ED(G) if and only if v is at maximum distance from u in G. A digraph G is said to be eccentric if there exists a digraph H such that G=ED(H). This paper is devoted to the study of the following two questions: what digraphs are eccentric and when the relation of being eccentric is symmetric. We present a characterization of eccentric digraphs, which in the undirected case says that a graph G is eccentric iff its complement graph G is either self-centered of radius two or it is the union of complete graphs. As a consequence, we obtain that all trees except those with diameter 3 are eccentric digraphs. We also determine when ED(G) is symmetric in the cases when G is a graph or a digraph that is not strongly connected. Crown Copyright © 2006 Published by Elsevier B.V. All rights reserved.
- Description: C1
- Description: 2003001601
On irregular total labellings
- Authors: Baca, Martin , Jendrol, Stanislav , Miller, Mirka , Ryan, Joe
- Date: 2007
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 307, no. 11-12 (May 2007), p. 1378-1388
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- Description: Two new graph characteristics, the total vertex irregularity strength and the total edge irregularity strength, are introduced. Estimations on these parameters are obtained. For some families of graphs the precise values of these parameters are proved. (c) 2006 Elsevier B.V. All rights reserved.
- Description: C1
- Description: 2003004909
Statistical limit inferior and limit superior for sequences of fuzzy numbers
- Authors: Aytar, Salih , Mammadov, Musa , Pehlivan, Serpil
- Date: 2006
- Type: Text , Journal article
- Relation: Fuzzy Sets and Systems Vol. 157, no. 7 (2006), p. 976-985
- Full Text: false
- Reviewed:
- Description: In this paper, we extend the concepts of statistical limit superior and limit inferior (as introduced by Fridy and Orhan [Statistical limit superior and limit inferior, Proc. Amer. Math. Soc. 125 (12) (1997) 3625-3631. [12]]) to statistically bounded sequences of fuzzy numbers and give some fuzzy-analogues of properties of statistical limit superior and limit inferior for sequences of real numbers. © 2005 Elsevier B.V. All rights reserved.
- Description: C1
- Description: 2003001832
Metric regularity and systems of generalized equations
- Authors: Dmitruk, Andrei , Kruger, Alexander
- Date: 2008
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 342, no. 2 (2008), p. 864-873
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- Description: The paper is devoted to a revision of the metric regularity property for mappings between metric or Banach spaces. Some new concepts are introduced: uniform metric regularity and metric multi-regularity for mappings into product spaces, when each component is perturbed independently. Regularity criteria are established based on a nonlocal version of Lyusternik-Graves theorem due to Milyutin. The criteria are applied to systems of generalized equations producing some "error bound" type estimates. © 2007 Elsevier Inc. All rights reserved.
Enumerations of vertex orders of almost Moore digraphs with selfrepeats
- Authors: Baskoro, Edy , Cholily, Yus Mochamad , Miller, Mirka
- Date: 2008
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 308, no. 1 (2008), p. 123-128
- Full Text: false
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- Description: An almost Moore digraph G of degree d > 1, diameter k > 1 is a diregular digraph with the number of vertices one less than the Moore bound. If G is an almost Moore digraph, then for each vertex u ∈ V (G) there exists a vertex v ∈ V (G), called repeat of u and denoted by r (u) = v, such that there are two walks of length ≤ k from u to v. The smallest positive integer p such that the composition rp (u) = u is called the order of u. If the order of u is 1 then u is called a selfrepeat. It is known that if G is an almost Moore digraph of diameter k ≥ 3 then G contains exactly k selfrepeats or none. In this paper, we propose an exact formula for the number of all vertex orders in an almost Moore digraph G containing selfrepeats, based on the vertex orders of the out-neighbours of any selfrepeat vertex. © 2007 Elsevier B.V. All rights reserved.
- Description: C1
Moore bound for mixed networks
- Authors: Nguyen, Minh Hoang , Miller, Mirka
- Date: 2008
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 308, no. 23 (Dec 2008), p. 5499-5503
- Full Text: false
- Reviewed:
- Description: Mixed graphs contain both undirected as well as directed links between vertices and therefore are an interesting model for interconnection communication networks. In this paper, we establish the Moore bound for mixed graphs, which generalizes both the directed and the undirected Moore bound. Crown Copyright (C) 2007 Published by Elsevier B.V. All rights reserved.
Colocality and twisted sums of Banach spaces
- Authors: Jebreen, H. M. , Jamjoom, F. B. H. , Yost, David
- Date: 2006
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 323, no. 2 (2006), p. 864-875
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- Description: Using the relation between subspaces of Banach spaces and quotients of their duals, we introduce the concept of colocality to give a new method that guarantees the existence of nontrivial twisted sums in which finite quotients play a major role (Theorem 1.7). An interesting point is that no restrictions are imposed on the quotients, only on the various subspaces. New examples of nontrivial twisted sums are given.
- Description: C1
- Description: 2003001831
New largest graphs of diameter 6. (Extended Abstract)
- Authors: Pineda-Villavicencio, Guillermo , Gomez, Jose , Miller, Mirka , Pérez-Rosés, Hebert
- Date: 2006
- Type: Text , Journal article
- Relation: Electronic Notes in Discrete Mathematics Vol. 24, no. (2006), p. 153-160
- Full Text:
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- Description: In the pursuit of obtaining largest graphs of given degree and diameter, many construction techniques have arisen. Compounding of graphs is one such technique. In this paper, by means of the compounding of complete graphs into the bipartite Moore graph of diameter 6, we obtain two families of (
- Description: C1
Structural properties of graphs of diameter 2 with maximal repeats
- Authors: Nguyen, Minh Hoang , Miller, Mirka
- Date: 2008
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 308, no. 11 (Jun 2008), p. 2337-2341
- Full Text: false
- Reviewed:
- Description: It was shown using eigenvalue analysis by Erdos et al. that with the exception of C-4, there are no graphs of diameter 2, of maximum degree d and of order d(2), that is, one less than the Moore bound. These graphs belong to a class of regular graphs of diameter 2, and having certain interesting structural properties, which will be proved in this paper. (c) 2007 Elsevier B.V. All rights reserved.
- Description: C1
Consecutive magic graphs
- Authors: Balbuena, Camino , Barker, Ewan , Lin, Yuqing , Miller, Mirka , Sugeng, Kiki Ariyanti
- Date: 2006
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 306, no. 16 (2006), p. 1817-1829
- Full Text: false
- Reviewed:
- Description: Let G be a graph of order n and size e. A vertex-magic total labeling is an assignment of the integers 1, 2, ..., n + e to the vertices and the edges of G, so that at each vertex, the vertex label and the labels on the edges incident at that vertex, add to a fixed constant, called the magic number of G. Such a labeling is a-vertex consecutive magic if the set of the labels of the vertices is { a + 1, a + 2, ..., a + n }, and is b-edge consecutive magic if the set of labels of the edges is { b + 1, b + 2, ..., b + e }. In this paper we prove that if an a-vertex consecutive magic graph has isolated vertices then the order and the size satisfy (n - 1)
- Description: C1
- Description: 2003001604
Diameter-sufficient conditions for a graph to be super-restricted connected
- Authors: Balbuena, Camino , Lin, Yuqing , Miller, Mirka
- Date: 2007
- Type: Text , Journal article
- Relation: Discrete Applied Mathematics Vol. , no. (2007), p.
- Full Text: false
- Reviewed:
- Description: A vertex-cut X is said to be a restricted cut of a graph G if it is a vertex-cut such that no vertex u in G has all its neighbors in X. Clearly, each connected component of G - X must have at least two vertices. The restricted connectivity
- Description: C1
Improved lower bound for the vertex connectivity of (delta;g)-cages
- Authors: Lin, Yuqing , Miller, Mirka , Balbuena, Camino
- Date: 2005
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 299, no. 1-3 (Aug 2005), p. 162-171
- Full Text: false
- Reviewed:
- Description: A (delta, g)-cage is a delta-regular graph with girth g and with the least possible number of vertices. We prove that all (delta, g)-cages are r-connected with r >= root(delta + 1) for g >= 7 odd. This result supports the conjecture of Fu, Huang and Rodger that all (delta; g)-cages are delta-connected. (c) 2005 Elsevier B.V. All rights reserved.
- Description: C1
- Description: 2003001397
On mixed Moore graphs
- Authors: Nguyen, Minh Hoang , Miller, Mirka , Gimbert, Joan
- Date: 2007
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 307, no. 7-8 (2007), p. 964-970
- Full Text: false
- Reviewed:
- Description: The Moore bound for a directed graph of maximum out-degree d and diameter k is M
- Description: C1
- Description: 2003005024
A sum labelling for the generalised friendship graph
- Authors: Fernau, Henning , Ryan, Joe , Sugeng, Kiki Ariyanti
- Date: 2008
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 308, no. 5-6 (2008), p. 734-740
- Full Text: false
- Reviewed:
- Description: We provide an optimal sum labelling scheme for the generalised friendship graph, also known as the flower (a symmetric collection of cycles meeting at a common vertex) and show that its sum number is 2. © 2007 Elsevier B.V. All rights reserved.
- Description: C1
Calculating the extremal number ex (v ; {C3, C4, ..., Cn})
- Authors: Tang, Jianmin , Lin, Yuqing , Miller, Mirka
- Date: 2006
- Type: Text , Journal article
- Relation: Electronic Notes in Discrete Mathematics Vol. 27, no. (2006), p. 101-102
- Full Text: false
- Reviewed:
- Description: This paper introduces and analyzes a parallel method of simulated annealing. Borrowing from genetic algorithms, an effective combination of simulated annealing and genetic algorithms, called parallel recombinative simulated annealing, is developed. This new algorithm strives to retain the desirable asymptotic convergence properties of simulated annealing, while adding the populations approach and recombinative power of genetic algorithms. The algorithm iterates a population of solutions rather than a single solution, employing a binary recombination operator as well as a unary neighborhood operator. Proofs of global convergence are given for two variations of the algorithm. Convergence behavior is examined, and empirical distributions are compared to Boltzmann distributions. Parallel recombinative simulated annealing is amenable to straightforward implementation on SIMD, MIMD, or shared-memory machines. The algorithm, implemented on the CM-5, is run repeatedly on two deceptive problems to demonstrate the added implicit parallelism and faster convergence which can result from larger population sizes.
- Description: C1
Necessary and sufficient conditions for stable conjugate duality
- Authors: Burachik, Regina , Jeyakumar, Vaithilingam , Wu, Zhiyou
- Date: 2006
- Type: Text , Journal article
- Relation: Journal of Nonlinear Analysis Vol. 64, no. 9 (2006), p. 1998-2005
- Full Text:
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- Description: The conjugate duality, which states that infx∈X φ(x, 0) = maxv∈Y ' −φ∗(0,v), whenever a regularity condition on φ is satisfied, is a key result in convex anal¬ysis and optimization, where φ : X × Y → IR ∪{+∞} is a convex function, X and Y are Banach spaces, Y ' is the continuous dual space of Y and φ∗ is the Fenchel-Moreau conjugate of φ. In this paper, we establish a necessary and sufficient condition for the stable conjugate duality, ∗ ∗ ∈ X' inf {φ(x, 0) + x ∗(x)} = max {−φ ∗(−x ,v)}, ∀x, x∈Xv∈Y ' and obtain a new global dual regularity condition, which is much more general than the popularly known interior-point type conditions, for the conjugate duality. As a consequence we present an epigraph closure condition which is necessary and sufficient for a stable Fenchel-Rockafellar duality theorem. In the case where one of the functions involved in the duality is a polyhedral convex function, we also provide generalized interior-point conditions for the epigraph closure condition. Moreover, we show that a stable Fenchel’s duality for sublinear functions holds whenever a subdifferential sum formula for the functions holds. As applications, we give general sufficient conditions for a minimax theorem, a subdifferential composition formula and for duality results of convex programming problems.
- Description: C1
- Description: 2003003596
Divisibility conditions in almost Moore digraphs with selfrepeats
- Authors: Teska, Jakub , Kuzel, Roman , Miller, Mirka
- Date: 2006
- Type: Text , Journal article
- Relation: Electronic Notes in Discrete Mathematics Vol. 24, no. (2006), p. 161-163
- Full Text: false
- Reviewed:
- Description: Moore digraph is a digraph with maximum out-degree d, diameter k and order Md, k = 1 + d + ... + dk. Moore digraphs exist only in trivial cases if d = 1 (i.e., directed cycle Ck) or k = 1 (i.e., complete symmetric digraph). Almost Moore digraphs are digraphs of order one less than Moore bound. We shall present new properties of almost Moore digraphs with selfrepeats from which we prove nonexistence of almost Moore digraphs for some k and d. © 2006 Elsevier B.V. All rights reserved.
- Description: C1
The core of a sequence of fuzzy numbers
- Authors: Aytar, Salih , Pehlivan, Serpil , Mammadov, Musa
- Date: 2008
- Type: Text , Journal article
- Relation: Fuzzy Sets and Systems Vol. 159, no. 24 (2008), p. 3369-3379
- Full Text: false
- Reviewed:
- Description: In this paper, based on level sets we define the limit inferior and limit superior of a bounded sequence of fuzzy numbers and prove some properties. We extend the concept of the core of a sequence of complex numbers, first introduced by Knopp in 1930, to a bounded sequence of fuzzy numbers and prove that the core of a sequence of fuzzy numbers is the interval [ν, μ] where ν and μ are extreme limit points of the sequence. © 2008 Elsevier B.V. All rights reserved.