Date of Award
College of Natual Science and Mathematics, Mathematics
Discrepancy inequality, Discrete curvature, Disjoint cycles, Routing number, Spectral graph theory
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.
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Received from ProQuest
Purcilly, Adam, "Discrepancy Inequalities in Graphs and Their Applications" (2020). Electronic Theses and Dissertations. 1824.