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Minimum Degree (minimum + degree)
Selected AbstractsNeurocognitive processes of the religious leader in ChristiansHUMAN BRAIN MAPPING, Issue 12 2009Jianqiao Ge Abstract Our recent work suggests that trait judgment of the self in Christians, relative to nonreligious subjects, is characterized by weakened neural coding of stimulus self-relatedness in the ventral medial prefrontal cortex (VMPFC) but enhanced evaluative processes of self-referential stimuli in the dorsal medial prefrontal cortex (DMPFC). The current study tested the hypothesis that Christian belief and practice produce a trait summary about the religious leader (Jesus) in the believers and thus episodic memory retrieval is involved to the minimum degree when making trait judgment of Jesus. Experiment 1 showed that to recall a specific incident to exemplify Jesus' trait facilitated behavioral performances associated with the following trait judgment of Jesus in nonreligious subjects but not in Christians. Experiment 2 showed that, for nonreligious subjects, trait judgments of both government and religious leaders resulted in enhanced functional connectivity between MPFC and posterior parietal cortex (PPC)/precuneus compared with self judgment. For Christian subjects, however, the functional connectivity between MPFC and PPC/precuneus differentiated between trait judgments of the government leader and the self but not between trait judgments of Jesus and the self. Our findings suggest that Christian belief and practice modulate the neurocognitive processes of the religious leader so that trait judgment of Jesus engages increased employment of semantic trait summary but decreased memory retrieval of behavioral episodes. Hum Brain Mapp, 2009. © 2009 Wiley-Liss, Inc. [source] Balanced judicious bipartitions of graphsJOURNAL OF GRAPH THEORY, Issue 3 2010Baogang Xu Abstract A bipartition of the vertex set of a graph is called balanced if the sizes of the sets in the bipartition differ by at most one. B. Bollobás and A. D. Scott, Random Struct Alg 21 (2002), 414,430 conjectured that if G is a graph with minimum degree of at least 2 then V(G) admits a balanced bipartition V1, V2 such that for each i, G has at most |E(G)|/3 edges with both ends in Vi. The minimum degree condition is necessary, and a result of B. Bollobás and A. D. Scott, J. Graph Theory 46 (2004), 131,143 shows that this conjecture holds for regular graphs G(i.e., when ,(G)=,(G)). We prove this conjecture for graphs G with ; hence, it holds for graphs ]ensuremathG with . © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 210,225, 2010 [source] Solution of a conjecture of Tewes and Volkmann regarding extendable cycles in in-tournamentsJOURNAL OF GRAPH THEORY, Issue 1 2010Dirk Meierling Abstract A directed cycle C of a digraph D is extendable if there exists a directed cycle C, in D that contains all vertices of C and an additional one. In 1989, Hendry defined a digraph D to be cycle extendable if it contains a directed cycle and every non-Hamiltonian directed cycle of D is extendable. Furthermore, D is fully cycle extendable if it is cycle extendable and every vertex of D belongs to a directed cycle of length three. In 2001, Tewes and Volkmann extended these definitions in considering only directed cycles whose length exceed a certain bound 3,k Total domination in 2-connected graphs and in graphs with no induced 6-cyclesJOURNAL OF GRAPH THEORY, Issue 1 2009Michael A. Henning Abstract A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number ,t(G) of G. It is known [J Graph Theory 35 (2000), 21,45] that if G is a connected graph of order n,>,10 with minimum degree at least 2, then ,t(G),,,4n/7 and the (infinite family of) graphs of large order that achieve equality in this bound are characterized. In this article, we improve this upper bound of 4n/7 for 2-connected graphs, as well as for connected graphs with no induced 6-cycle. We prove that if G is a 2-connected graph of order n,>,18, then ,t(G),,,6n/11. Our proof is an interplay between graph theory and transversals in hypergraphs. We also prove that if G is a connected graph of order n,>,18 with minimum degree at least 2 and no induced 6-cycle, then ,t(G),,,6n/11. Both bounds are shown to be sharp. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 55,79, 2009 [source] On a conjecture about edge irregular total labelingsJOURNAL OF GRAPH THEORY, Issue 4 2008Stephan Brandt Abstract As our main result, we prove that for every multigraph G,=,(V, E) which has no loops and is of order n, size m, and maximum degree there is a mapping such that for every with . Functions with this property were recently introduced and studied by Ba,a et al. and were called edge irregular total labelings. Our result confirms a recent conjecture of Ivan,o and Jendrol, about such labelings for dense graphs, for graphs where the maximum and minimum degree are not too different in terms of the order, and also for large graphs of bounded maximum degree. © 2008 Wiley Periodicals, Inc. J Graph Theory 57: 333,343, 2008 [source] On k -domination and minimum degree in graphsJOURNAL OF GRAPH THEORY, Issue 1 2008Odile Favaron Abstract A subset S of vertices of a graph G is k -dominating if every vertex not in S has at least k neighbors in S. The k -domination number is the minimum cardinality of a k -dominating set of G. Different upper bounds on are known in terms of the order n and the minimum degree of G. In this self-contained article, we present an Erdös-type result, from which some of these bounds follow. In particular, we improve the bound for , proved by Chen and Zhou in 1998. Furthermore, we characterize the extremal graphs in the inequality , if , of Cockayne et al. This characterization generalizes that of graphs realizing . © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 33,40, 2008 [source] Forcing highly connected subgraphsJOURNAL OF GRAPH THEORY, Issue 4 2007Maya Jakobine Stein Abstract A theorem of Mader states that highly connected subgraphs can be forced in finite graphs by assuming a high minimum degree. We extend this result to infinite graphs. Here, it is necessary to require not only high degree for the vertices but also high vertex-degree (or multiplicity) for the ends of the graph, that is, a large number of disjoint rays in each end. We give a lower bound on the degree of vertices and the vertex-degree of the ends which is quadratic in k, the connectedness of the desired subgraph. In fact, this is not far from best possible: we exhibit a family of graphs with a degree of order 2k at the vertices and a vertex-degree of order k log k at the ends which have no k -connected subgraphs. Furthermore, if in addition to the high degrees at the vertices, we only require high edge-degree for the ends (which is defined as the maximum number of edge-disjoint rays in an end), Mader's theorem does not extend to infinite graphs, not even to locally finite ones. We give a counterexample in this respect. But, assuming a lower bound of at least 2k for the edge-degree at the ends and the degree at the vertices does suffice to ensure the existence (k + 1)- edge -connected subgraphs in arbitrary graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 331,349, 2007 [source] Sufficient conditions for graphs to be ,,-optimal, super-edge-connected, and maximally edge-connectedJOURNAL OF GRAPH THEORY, Issue 3 2005Angelika Hellwig Abstract The restricted-edge-connectivity of a graph G, denoted by ,,(G), is defined as the minimum cardinality over all edge-cuts S of G, where G - S contains no isolated vertices. The graph G is called ,,-optimal, if ,,(G),=,,(G), where ,(G) is the minimum edge-degree in G. A graph is super-edge-connected, if every minimum edge-cut consists of edges adjacent to a vertex of minimum degree. In this paper, we present sufficient conditions for arbitrary, triangle-free, and bipartite graphs to be ,,-optimal, as well as conditions depending on the clique number. These conditions imply super-edge-connectivity, if , (G),,,3, and the equality of edge-connectivity and minimum degree. Different examples will show that these conditions are best possible and independent of other results in this area. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 228,246, 2005 [source] Some remarks on dominationJOURNAL OF GRAPH THEORY, Issue 3 2004D. Archdeacon Abstract We prove a conjecture of Favaron et al. that every graph of order n and minimum degree at least three has a total dominating set of size at least n/2. We also present several related results about: (1) extentions to graphs of minimum degree two, (2) examining graphs where the bound is tight, and (3) a type of bipartite domination and its relation to transversals in hypergraphs. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 207,210, 2004 [source] Toughness, minimum degree, and spanning cubic subgraphsJOURNAL OF GRAPH THEORY, Issue 2 2004D. Bauer Abstract Degree conditions on the vertices of a t -tough graph G (1,,,t,<,3) are presented which ensure the existence of a spanning cubic subgraph in G. These conditions are best possible to within a small additive constant for every fixed rational t ,[1,4/3),[2,8/3). © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 119,141, 2004 [source] Degree sequence conditions for equal edge-connectivity and minimum degree, depending on the clique numberJOURNAL OF GRAPH THEORY, Issue 3 2003Lutz Volkmann Abstract Using the well-known Theorem of Turán, we present in this paper degree sequence conditions for the equality of edge-connectivity and minimum degree, depending on the clique number of a graph. Different examples will show that these conditions are best possible and independent of all the known results in this area. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 234,245, 2003 [source] On removable cycles through every edgeJOURNAL OF GRAPH THEORY, Issue 2 2003Manoel Lemos Abstract Mader and Jackson independently proved that every 2-connected simple graph G with minimum degree at least four has a removable cycle, that is, a cycle C such that G/E(C) is 2-connected. This paper considers the problem of determining when every edge of a 2-connected graph G, simple or not, can be guaranteed to lie in some removable cycle. The main result establishes that if every deletion of two edges from G remains 2-connected, then, not only is every edge in a removable cycle but, for every two edges, there are edge-disjoint removable cycles such that each contains one of the distinguished edges. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 155,164, 2003 [source] A generalization of an edge-connectivity theorem of ChartrandNETWORKS: AN INTERNATIONAL JOURNAL, Issue 2 2009Frank Boesch Abstract In 1966, Chartrand proved that if the minimum degree of a graph is at least the floor of half the number of nodes, then its edge-connectivity equals its minimum degree. A more discriminating notion of edge-connectivity is introduced, called the k -component order edge-connectivity, which is the minimum number of edges required to be removed so that the order of each component of the resulting subgraph is less than k. Results are established that guarantee that this parameter is at least as large as the minimum degree, provided the minimum degree is sufficiently large. This generalizes Chartrand's result. It is also determined when these results are best possible. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009 [source] Sufficient conditions for a graph to be super restricted edge-connectedNETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2008Shiying Wang Abstract Restricted edge connectivity is a more refined network reliability index than edge connectivity. A restricted edge cut F of a connected graph G is an edge cut such that G - F has no isolated vertex. The restricted edge connectivity ,, is the minimum cardinality over all restricted edge cuts. We call G ,,-optimal if ,, = ,, where , is the minimum edge degree in G. Moreover, a ,,-optimal graph G is called a super restricted edge-connected graph if every minimum restricted edge cut separates exactly one edge. Let D and g denote the diameter and girth of G, respectively. In this paper, we first present a necessary condition for non-super restricted edge-connected graphs with minimum degree , , 3 and D , g , 2. Next, we prove that a connected graph with minimum degree , , 3 and D , g , 3 is super restricted edge-connected. Finally, we give some sufficient conditions on the conditional diameter and the girth for super restricted edge-connected graphs. © 2007 Wiley Periodicals, Inc. NETWORKS, 2008 [source] Connectedness of digraphs and graphs under constraints on the conditional diameterNETWORKS: AN INTERNATIONAL JOURNAL, Issue 2 2005X. Marcote Abstract Given a digraph G with minimum degree , and an integer 0, , , ,, consider every pair of vertex subsets V1 and V2 such that both the minimum out-degree of the induced subdigraph G[V1] and the minimum in-degree of G[V2] are at least ,. The conditional diameter D, of G is defined as the maximum of the distances d(V1, V2) between any two such vertex subsets. Clearly, D0 is the standard diameter and D0 , D1 , ··· , D, holds. In this article, we guarantee appropriate lower bounds for the connectivities and superconnectivities of a digraph G when D, , h(,,), h(,,) being a function of the parameter ,,,which is related to the shortest paths in G. As a corollary of these results, we give some constraints of the kind D, , h(,,), which assure that the digraph is maximally connected, maximally edge-connected, superconnected, or edge-superconnected, extending other previous results of the same kind. Similar statements can be obtained for a graph as a direct consequence of those for its associated symmetric digraph. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45(2), 80,87 2005 [source] Approximation algorithms for finding low-degree subgraphsNETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2004Philip N. Klein Abstract We give quasipolynomial-time approximation algorithms for designing networks with a minimum degree. Using our methods, one can design networks whose connectivity is specified by "proper" functions, a class of 0,1 functions indicating the number of edges crossing each cut. We also provide quasipolynomial-time approximation algorithms for finding two-edge-connected spanning subgraphs of approximately minimum degree of a given two-edge-connected graph, and a spanning tree (branching) of approximately minimum degree of a directed graph. The degree of the output network in all cases is guaranteed to be at most (1 + ,) times the optimal degree, plus an additive O(log1+,n) for any , > 0. Our analysis indicates that the degree of an optimal subgraph for each of the problems above is well estimated by certain polynomially solvable linear programs. This suggests that the linear programs we describe could be useful in obtaining optimal solutions via branch and bound. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(3), 203,215 2004 [source] Edge-superconnectivity of cagesNETWORKS: AN INTERNATIONAL JOURNAL, Issue 1 2004X. Marcote Abstract A graph with minimum degree , is said to be edge-superconnected if each minimum edge-cut consists of all the edges incident with some vertex (so , = ,). A smallest ,-regular graph G with girth g is said to be a (,, g)-cage. We show that every (,, g)-cage with odd girth is edge-superconnected. This result strengthens one obtained by Wang et al. (, = , for every such cage) and supports the conjecture of Fu et al. that all (,, g)-cages are ,-connected. © 2003 Wiley Periodicals, Inc. [source] Superconnected digraphs and graphs with small conditional diametersNETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2002C. Balbuena Abstract The conditional diameter D, of a digraph G measures how far apart a pair of vertex sets V1 and V2 can be in such a way that the minimum out-degree and the minimum in-degree of the subdigraphs induced by V1 and V2, respectively, are at least ,. Thus, D0 is the standard diameter and D0 , D1 , ··· , D,, where , is the minimum degree. We prove that if D, , 2l , 3, where l is a parameter related to the shortest paths, then G is maximally connected, is superconnected, or has a good superconnectivity, depending only on whether , is equal to ,,/2,, ,(, , 1)/2,, or ,(, , 1)/3,, respectively. In the edge case, it is enough that D, , 2l , 2. The results for graphs are obtained as a corollary of those for digraphs, because, in the undirected case, l = ,(g , 1)/2,, g being the girth. © 2002 Wiley Periodicals, Inc. [source] Tailoring viscoelastic and mechanical properties of the foamed blends of EVA and various ethylene-styrene interpolymersPOLYMER COMPOSITES, Issue 3 2003I-Chun Liu Foamed materials (EVA/ESI) have been prepared from blends of ethylene-vinyl acetate copolymer (EVA) and ethylene-styrene interpolymers (ESI) in the presence of various amounts of dicumyl peroxide (DCP). Four ESIs of different compositions were employed in this study; their styrene contents ranged from 30 to 73 wt% and their Tg ranged from ,2 to 33°C. It has been found that microcellular morphology, degree of crosslinking and expansion ratio were strongly affected by the DCP concentration and the type of ESI employed. A minimum degree of crosslinking was required for making good foams and the same degree of crosslinking could be achieved by employing a smaller amount of DCP for an EVA/ESI blend having a higher styrene content. In contrast to other EVA blends, such as EVA/LDPE, these EVA/ESI blends exhibited no existence of any optimum DCP concentration, and the , glass transition temperatures of the foams varied with the ESI type, covering a wide span from 0°C to 37°C. Therefore, it was possible to tailor the Tg of an EVA/ESI blend by choosing an appropriate type of ESI. Furthermore, by correctly tailoring the Tg, the EVA/ESI foam could be made into a rubbery material with a custom-designed damping factor. Tensile strength and modulus of the EVA/ESI foams increased generally with an increase in the styrene content, with the exception that ESIs with very low styrene content will confer on the blend a high modulus at small strain and a large elongation at break. [source] An optimization procedure for the pultrusion process based on a finite element formulationPOLYMER COMPOSITES, Issue 3 2002R. M. L. Coelho Composite materials are manufactured by different processes. In all, the process variables have to be analyzed in order to obtain a part with uniform mechanical properties. In the pultrusion process, two variables are the most important: the pulling speed of resin-impregnated fibers and the temperature profile (boundary condition) imposed on the mold wall. Mathematical modeling of this process results in partial differential equations that are solved here by a detailed procedure based on the Galerkin weighted residual finite element method. The combination of the Picard and Newton-Raphson methods with an analytical Jacobian calculation proves to be robust, and a mesh adaptation procedure is presented in order to avoid integration errors during the process optimization. The two earlier-mentioned variables are optimized by the Simulated Annealing method with some constraints, such as a minimum degree of cure at the end of the process, and the resin degradation (the part temperature cannot be higher than the resin degradation temperature at any time during the whole process). Herein, the proposed objective function is an economic criterion instead of the pulling speed of resin-impregnated fibers, used in the majority of papers. [source] Proof of a tiling conjecture of KomlósRANDOM STRUCTURES AND ALGORITHMS, Issue 2 2003Ali Shokoufandeh Abstract A conjecture of Komlós states that for every graph H, there is a constant K such that if G is any n -vertex graph of minimum degree at least (1 , (1/,cr(H)))n, where ,cr(H) denotes the critical chromatic number of H, then G contains an H -matching that covers all but at most K vertices of G. In this paper we prove that the conjecture holds for all sufficiently large values of n. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 180,205, 2003 [source]
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