Discrepancy Inequalities in Graphs and Their Applications

Spectral graph theory, which is the use of eigenvalues of matrices associated with graphs, is a modern technique that has expanded our understanding of graphs and their structure. A particularly useful tool in spectral graph theory is the Expander Mixing Lemma, also known as the discrepancy inequality, which bounds the edge distribution between two sets based on the spectral gap. More specifically, it states that a small spectral gap of a graph implies that the edge distribution is close to random. This dissertation uses this tool to study two problems in extremal graph theory, then produces similar discrepancy inequalities based not on the spectral gap of a graph, but rather a different tool with motivations in Riemannian geometry. The first problem explored in this dissertation is motivated by parallel computing and other communication networks. Consider a connected graph G, with a pebble placed on each vertex of G. The routing number, rt(G), of G is the minimum number of steps needed to route any permutation on the vertices of G, where a step consists of selecting a matching in the graph and swapping the pebbles on the endpoints of each edge. Alon, Chung, and Graham introduced this parameter, and (among other results) gave a bound based on the spectral gap for general graphs. The bound they obtain is polylogarithmic for graphs with a sufficiently strong spectral gap. In this dissertation, we use the Expander Mixing Lemma, the probablistic method, and other extremal tools to investigate when this upper bound can be improved to be constant depending on the gap and the vertex degrees. The second problem examined in this dissertation has motivations in a question of Erdõs and Pósa, who conjectured that every sufficiently dense graph on n vertices, where n is divisible by 3, decomposes into triangles. While Corradi and Hajnal proved this result true for graphs with minimum degree at least (2/3)n, their result spawned a series of similar questions about the number of vertex-disjoint subgraphs of a certain class that a graph with some degree condition must contain. While this problem is well-studied for dense graphs, many results give significantly worse bounds for less dense graphs. Using spectral graph theory, we show that every graph with some weak density and spectral conditions contains O(sqrt(nd)) vertex-disjoint cycles. Furthermore, even if we require these cycles to contain a certain number of chords, a graph satisfying these conditions will still contain O(sqrt(nd)) such vertex-disjoint cycles. In both cases, we show this bound to be best possible. Finally, we conclude by obtaining local version of a discrepancy inequality. An oversimplification of the Expander Mixing Lemma states that a graph with a strong spectral condition must have nice edge distribution. We seek to mimic that idea, but by using discrete curvature instead of a spectral condition. Discrete curvature, inspired by its counterpart in Riemannian geometry, measures the local volume growth at a vertex. Thus, given a vertex x, our result uses curvature to quantify the edge distribution between vertices that are a distance one from x and vertices that are a distance two from x. In doing this, we are able to study the number of 3-cycles and 4-cycles containing a particular edge

Extremal problems on cycle structure and colorings of graphs

In this Thesis, we consider two main themes: conditions that guarantee diverse cycle structure within a graph, and the existence of strong edge-colorings for a specific family of graphs. In Chapter 2 we consider a question closely related to the Matthews-Sumner conjecture, which states that every 4-connected claw-free graph is Hamiltonian. Since there exists an infinite family of 4-connected claw-free graphs that are not pancyclic, Gould posed the problem of characterizing the pairs of graphs, {X,Y}, such that every 4-connected {X,Y}-free graph is pancyclic. In this chapter we describe a family of pairs of graphs such that if every 4-connected {X,Y}-free graph is pancyclic, then {X,Y} is in this family. Furthermore, we show that every 4-connected {K_(1,3),N(4,1,1)}-free graph is pancyclic. This result, together with several others, completes a characterization of the family of subgraphs, F such that for all H in ∈, every 4-connected {K_(1,3), H}-free graph is pancyclic. In Chapters and 4 we consider refinements of results on cycles and chorded cycles. In 1963, Corrádi and Hajnal proved a conjecture of Erdös, showing that every graph G on at least 3k vertices with minimum degree at least 2k contains k disjoint cycles. This result was extended by Enomoto and Wang, who independently proved that graphs on at least 3kvertices with minimum degree-sum at least 4k - 1 also contain k disjoint cycles. Both results are best possible, and recently, Kierstead, Kostochka, Molla, and Yeager characterized their sharpness examples. A chorded cycle analogue to the result of Corrádi and Hajnal was proved by Finkel, and a similar analogue to the result of Enomoto and Wang was proved by Chiba, Fujita, Gao, and Li. In Chapter 3 we characterize the sharpness examples to these statements, which provides a chorded cycle analogue to the characterization of Kierstead et al. In Chapter 4 we consider another result of Chiba et al., which states that for all integers r and s with r + s ≥ 1, every graph G on at least 3r + 4s vertices with ẟ(G) ≥ 2r+3s contains r disjoint cycles and s disjoint chorded cycles. We provide a characterization of the sharpness examples to this result, which yields a transition between the characterization of Kierstead et al. and the main result of Chapter 3. In Chapter 5 we move to the topic of edge-colorings, considering a variation known as strong edge-coloring. In 1990, Faudree, Gyárfás, Schelp, and Tuza posed several conjectures regarding strong edge-colorings of subcubic graphs. In particular, they conjectured that every subcubic planar graph has a strong edge-coloring using at most nine colors. We prove a slightly stronger form of this conjecture, showing that it holds for all subcubic planar loopless multigraphs

Brushless d.c. torque motor first quarterly report, 25 jun. - 25 sep. 1964

Brushless direct current torque moto

Final report for a brushless dc torque motor

Brushless direct current torque motor using permanent magnet rotor and three-phase winding in stationary armature for operation in vacuu

On the use of generating functions for topics in clustered networks

In this thesis we relax the locally tree-like assumption of configuration model random networks to examine the properties of clustering, and the effects thereof, on bond percolation. We introduce an algorithmic enumeration method to evaluate the probability that a vertex remains unattached to the giant connected component during percolation. The properties of the non-giant, finite components of clustered networks are also examined, along with the degree correlations between subgraphs. In a second avenue of research, we investigate the role of clustering on 2-strain epidemic processes under various disease interaction schedules. We then examine an -generation epidemic by performing repeated percolation events

Modelling of rotor defects in squirrel cage induction motors using the time stepping numerical field analysis

The arrival of cheap and fast microcomputers has stimulated the development of machine condition monitoring systems. Such systems measure one or more machine performance parameters, with a view to detecting the early signs of failure and initiating some form of action. To help developing these systems, a thorough knowledge of the behaviour of the machine in its post-fault condition must be well known and well understood. With this view in mind, a simple, yet reasonably accurate method of analysing the induction machine magnetic circuit under normal and abnormal conditions at moderate cost has been developed. Numerical field analysis is applied to the calculation of induction motors. It is based on a very simple Magnetic Network Technique (MNT) solution of the magnetic field. The field is assumed to be two-dimensional. The three dimensionality of the machine is taken into account within the two dimensional model. The general time-dependence of the field and the motion of the rotor are modelled correctly in a step-by-step solution. The model uses relatively small amount of computation time when compared with the previous methods of analysis. This technique is used for the evaluation of the broken bars effects on the machines performance and magnetic field variation. A series of experimental tests have been carried out. The results obtained are directly compared with the computed ones and they showed a good correlation. Finally, suggestions for suitable methods for the detection of broken bars are given along with some suggestions for future work