Orthogonal polarity graphs and Sidon sets

Determining the maximum number of edges in an $n$-vertex $C_4$-free graph is a well-studied problem that dates back to a paper of Erd\H{o}s from 1938. One of the most important families of $C_4$-free graphs are the Erd\H{o}s-R\'enyi orthogonal polarity graphs. We show that the Cayley sum graph constructed using a Bose-Chowla Sidon set is isomorphic to a large induced subgraph of the Erd\H{o}s-R\'enyi orthogonal polarity graph. Using this isomorphism we prove that the Petersen graph is a subgraph of every sufficiently large Erd\H{o}s-R\'enyi orthogonal polarity graph.Comment: The authors would like to thank Jason Williford for noticing an error in the proof of Theorem 1.2 in the previous version. This error has now been correcte

EXPLORATIONS IN HOMEOVISCOUS ADAPTATION AND MASS SPECTRAL ANALYSIS OF MEMBRANE LIPIDS

The focus of this dissertation is centered on the mass spectral analysis of lipids and changes occurring in keeping with the concept of homeoviscous adaptation [1]. Homeoviscous adaptation is the process of modification of membrane lipids in response to environmental stimuli [1]. Dissertation investigations applied this concept to prokaryotic and eukaryotic organisms, and expanded the perception of environmental factors from exogenous organic solvents to intracellular environment. The field of lipidomics deals with the analysis of phospholipid and fatty acid components of membranes the changes that occur due to environmental stimuli and their biological significance [2-6]. The high sensitivity of mass spectrometry (MS) is an ideal tool for lipidomics allowing detection, quantification and structural elucidation [6]. Coupling of a mass spectrometer to a chromatographic system, such as gas chromatograph (GC), allows the separation of fatty acid methyl esters analytes prior to analysis [7]. The research investigations that comprise this dissertation are divided into three interrelated projects. The first project involved the analysis of composition and structure of Clostridium thermocellum membranes from wild-type and ethanol-adapted strains in response to adaptation of cultures to growth in ethanol. The hypothesis being that adaptation of cultures to growth in ethanol would result in compensatory change to the membrane composition. Rat mitochondrial fatty acid profiles isolated from brain, liver, kidney and heart tissues were compared. The hypothesis being that differences in cellular environments found among various tissues would be reflected in the mitochondrial membrane composition. These data support the concept that variations to the lipid content of neurological mitochondria may increase susceptibility to the products of oxidative stress. Lastly, changes in neurological mitochondria as a function of Alzheimer’s disease progression were studied. The hypothesis being that changes to the mitochondrial lipidome would be significantly reflected during advanced stages of AD, in addition to being more prevalent in regions displaying greater pathology. The three interrelated projects increased our understanding of the boundaries established by the concept of homeoviscous adaptation. Project specific hypotheses were supported by data obtained from these investigations

Forbidden subgraphs and complete partitions

A graph is called an $(r,k)$-graph if its vertex set can be partitioned into $r$ parts of size at most $k$ with at least one edge between any two parts. Let $f(r,H)$ be the minimum $k$ for which there exists an $H$-free $(r,k)$-graph. In this paper we build on the work of Axenovich and Martin, obtaining improved bounds on this function when $H$ is a complete bipartite graph, even cycle, or tree. Some of these bounds are best possible up to a constant factor and confirm a conjecture of Axenovich and Martin in several cases. We also generalize this extremal problem to uniform hypergraphs and prove some initial results in that setting