Weighted degrees and heavy cycles in weighted graphs

AbstractA weighted graph is a graph provided with an edge-weighting function w from the edge set to nonnegative real numbers. Bondy and Fan [Annals of Discrete Math. 41 (1989), 53–69] began the study on the existence of heavy cycles in weighted graphs. Though several results with Dirac-type degree condition can be generalized to an Ore-type one in unweighted graphs, it is shown in Bondy et al. [Discuss. Math. Graph Theory 22 (2002), 7–15] that Bondy and Fan’s theorem, which uses Dirac-type condition, cannot be generalized analogously by using Ore-type condition.In this paper we investigate the property peculiar to weighted graphs, and prove a theorem on the existence of heavy cycles in weighted graphs under an Ore-type condition, which generalizes Bondy and Fan’s theorem. Moreover, we show the existence of heavy cycles passing through some specified vertices

Analyzing and Modeling Real-World Phenomena with Complex Networks: A Survey of Applications

The success of new scientific areas can be assessed by their potential for contributing to new theoretical approaches and in applications to real-world problems. Complex networks have fared extremely well in both of these aspects, with their sound theoretical basis developed over the years and with a variety of applications. In this survey, we analyze the applications of complex networks to real-world problems and data, with emphasis in representation, analysis and modeling, after an introduction to the main concepts and models. A diversity of phenomena are surveyed, which may be classified into no less than 22 areas, providing a clear indication of the impact of the field of complex networks.Comment: 103 pages, 3 figures and 7 tables. A working manuscript, suggestions are welcome

Hamiltonicity, Pancyclicity, and Cycle Extendability in Graphs

The study of cycles, particularly Hamiltonian cycles, is very important in many applications. Bondy posited his famous metaconjecture, that every condition sufficient for Hamiltonicity actually guarantees a graph is pancyclic. Pancyclicity is a stronger structural property than Hamiltonicity. An even stronger structural property is for a graph to be cycle extendable. Hendry conjectured that any graph which is Hamiltonian and chordal is cycle extendable. In this dissertation, cycle extendability is investigated and generalized. It is proved that chordal 2-connected K1,3-free graphs are cycle extendable. S-cycle extendability was defined by Beasley and Brown, where S is any set of positive integers. A conjecture is presented that Hamiltonian chordal graphs are {1, 2}-cycle extendable. Dirac’s Theorem is an classic result establishing a minimum degree condition for a graph to be Hamiltonian. Ore’s condition is another early result giving a sufficient condition for Hamiltonicity. In this dissertation, generalizations of Dirac’s and Ore’s Theorems are presented. The Chvatal-Erdos condition is a result showing that if the maximum size of an independent set in a graph G is less than or equal to the minimum number of vertices whose deletion increases the number of components of G, then G is Hamiltonian. It is proved here that the Chvatal-Erdos condition guarantees that a graph is cycle extendable. It is also shown that a graph having a Hamiltonian elimination ordering is cycle extendable. The existence of Hamiltonian cycles which avoid sets of edges of a certain size and certain subgraphs is a new topic recently investigated by Harlan, et al., which clearly has applications to scheduling and communication networks among other things. The theory is extended here to bipartite graphs. Specifically, the conditions for the existence of a Hamiltonian cycle that avoids edges, or some subgraph of a certain size, are determined for the bipartite case. Briefly, this dissertation contributes to the state of the art of Hamiltonian cycles, cycle extendability and edge and graph avoiding Hamiltonian cycles, which is an important area of graph theory

Micro-, Meso- and Macro-Connectomics of the Brain

Neurosciences, Neurolog

Self-Evaluation Applied Mathematics 2003-2008 University of Twente

This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008

Resource Allocation Schemes And Performance Evaluation Models For Wavelength Division Multiplexed Optical Networks

Wavelength division multiplexed (WDM) optical networks are rapidly becoming the technology of choice in network infrastructure and next-generation Internet architectures. WDM networks have the potential to provide unprecedented bandwidth, reduce processing cost, achieve protocol transparency, and enable efficient failure handling. This dissertation addresses the important issues of improving the performance and enhancing the reliability of WDM networks as well as modeling and evaluating the performance of these networks. Optical wavelength conversion is one of the emerging WDM enabling technologies that can significantly improve bandwidth utilization in optical networks. A new approach for the sparse placement of full wavelength converters based on the concept of the k-Dominating Set (k-DS) of a graph is presented. The k-DS approach is also extended to the case of limited conversion capability using three scalable and cost-effective switch designs: flexible node-sharing, strict node-sharing and static mapping. Compared to full search algorithms previously proposed in the literature, the K-DS approach has better blocking performance, has better time complexity and avoids the local minimum problem. The performance benefit of the K-DS approach is demonstrated by extensive simulation. Fiber delay line (FDL) is another emerging WDM technology that can be used to obtain limited optical buffering capability. A placement algorithm, k-WDS, for the sparse placement of FDLs at a set of selected nodes in Optical Burst Switching (OBS) networks is proposed. The algorithm can handle both uniform and non-uniform traffic patterns. Extensive performance tests have shown that k-WDS provides more efficient placement of optical fiber delay lines than the well-known approach of placing the resources at nodes with the highest experienced burst loss. Performance results that compare the benefit of using FDLs versus using optical wavelength converters (OWCs) are presented. A new algorithm, A-WDS, for the placement of an arbitrary numbers of FDLs and OWCs is introduced and is evaluated under different non-uniform traffic loads. This dissertation also introduces a new cost-effective optical switch design using FDL and a QoS-enhanced JET (just enough time) protocol suitable for optical burst switched WDM networks. The enhanced JET protocol allows classes of traffic to benefit from FDLs and OWCs while minimizing the end-to-end delay for high priority bursts. Performance evaluation models of WDM networks represent an important research area that has received increased attention. A new analytical model that captures link dependencies in all-optical WDM networks under uniform traffic is presented. The model enables the estimation of connection blocking probabilities more accurately than previously possible. The basic formula of the dependency between two links in this model reflects their degree of adjacency, the degree of connectivity of the nodes composing them and their carried traffic. The usefulness of the model is illustrated by applying it to the sparse wavelength converters placement problem in WDM networks. A lightpath containing converters is divided into smaller sub-paths such that each sub-path is a wavelength continuous path and the nodes shared between these sub-paths are full wavelength conversion capable. The blocking probability of the entire path is obtained by computing the blocking probabilities of the individual sub-paths. The analytical-based sparse placement algorithm is validated by comparing it with its simulation-based counterpart using a number of network topologies. Rapid recovery from failure and high levels of reliability are extremely important in WDM networks. A new Fault Tolerant Path Protection scheme, FTPP, for WDM mesh networks based on the alarming state of network nodes and links is introduced. The results of extensive simulation tests show that FTPP outperforms known path protection schemes in terms of loss of service ratio and network throughput. The simulation tests used a wide range of values for the load intensity, the failure arrival rate and the failure holding time. The FTPP scheme is next extended to the differentiated services model and its connection blocking performance is evaluated. Finally, a QoS-enhanced FTPP (QEFTPP) routing and path protection scheme in WDM networks is presented. QEFTPP uses preemption to minimize the connection blocking percentage for high priority traffic. Extensive simulation results have shown that QEFTPP achieves a clear QoS differentiation among the traffic classes and provides a good overall network performance

EUSN 2021 Book of Abstracts, Fifth European Conference on Social Networks

Book of abstract of the fifth European conference on Social Networks EUSN 202

Gonality of metric graphs and Catalan-many tropical morphisms to trees

This thesis consists of two points of view to regard degree-(g′+1) tropical morphisms Φ : (Γ,w) → Δ from a genus-(2g′) weighted metric graph (Γ,w) to a metric tree Δ, where g′ is a positive integer. The first point of view, developed in Part I, is purely combinatorial and constructive. It culminates with an application to bound the gonality of (Γ,w). The second point of view, developed in Part II, incorporates category theory to construct a unified framework under which both Φ and higher dimensional analogues can be understood. These higher dimensional analogues appear in the construction of a moduli space Gtrop/g→0,d parametrizing the tropical morphisms Φ, and a moduli spaceMtrop/g parametrizing the (Γ,w). There is a natural projection map Π : Gtrop/g→0,d →Mtrop/g that sends Φ : (Γ,w) → Δ to (Γ,w). The strikingly beautiful result is that when g = 2g′ and d = g′+1, the projection Π itself is an indexed branched cover, thus having the same nature as the maps Φ that are being parametrized. Moreover, fibres of Π have Catalan-many points. Each part has its own introduction that motivates and describes the problem from its own perspective. Part I and its introduction are based on two articles which are joint work with Jan Draisma. Part II contains material intended to be published as two articles. There is also a layman summary available at the beginning