Cyclic orthogonal double covers of 6-regular circulant graphs by disconnected forests

An orthogonal double cover (ODC) of a graph H is a collection G = {Gv : v ∈ V (H)} of |V (H)| subgraphs of H such that every edge of H is contained in exactly two members of G and for any two members Gu and Gv in G, |E(Gu) ∩ E(Gv)| is 1 if u and v are adjacent in H and it is 0 if u and v are nonadjacent in H. An ODC G of H is cyclic if the cyclic group of order |V (H)| is a subgroup of the automorphism group of G; otherwise it is noncyclic. Recently, Sampathkumar and Srinivasan settled the problem of the existence of cyclic ODCs of 4-regular circulant graphs. An ODC G of H is cyclic (CODC) if the cyclic group of order | V (H)| is a subgroup of the automorphism group of G, the set of all automorphisms of G; otherwise it is noncyclic. In this paper, we have completely settled the existence problem of CODCs of 6-regular circulant graphs by four acyclic disconnected graphs.Publisher's Versio

Embedding rainbow trees with applications to graph labelling and decomposition

A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares. Since then rainbow structures have been the focus of extensive research and have found applications in the areas of graph labelling and decomposition. An edge-colouring is locally k-bounded if each vertex is contained in at most k edges of the same colour. In this paper we prove that any such edge-colouring of the complete graph Kn contains a rainbow copy of every tree with at most (1−o(1))n/k vertices. As a locally k-bounded edge-colouring of Kn may have only (n−1)/k distinct colours, this is essentially tight. As a corollary of this result we obtain asymptotic versions of two long-standing conjectures in graph theory. Firstly, we prove an asymptotic version of Ringel's conjecture from 1963, showing that any n-edge tree packs into the complete graph K(2n+o(n)) to cover all but o(n^2) of its edges. Secondly, we show that all trees have an almost-harmonious labelling. The existence of such a labelling was conjectured by Graham and Sloane in 1980. We also discuss some additional applications

Some Graph Laplacians and Variational Methods Applied to Partial Differential Equations on Graphs

In this dissertation we will be examining partial differential equations on graphs. We start by presenting some basic graph theory topics and graph Laplacians with some minor original results. We move on to computing original Jost graph Laplacians of friendly labelings of various finite graphs. We then continue on to a host of original variational problems on a finite graph. The first variational problem is an original basic minimization problem. Next, we use the Lagrange multiplier approach to the Kazdan-Warner equation on a finite graph, our original results generalize those of Dr. Grigor’yan, Dr. Yang, and Dr. Lin. Then we do an original saddle point approach to the Ahmad, Lazer, and Paul resonant problem on a finite graph. Finally, we tackle an original Schrödinger operator variational problem on a locally finite graph inspired by some papers written by Dr. Zhang and Dr. Pankov. The main keys to handling this difficult breakthrough Schrödinger problem on a locally finite graph are Dr. Costa’s definition of uniformly locally finite graph and the locally finite graph analog Dr. Zhang and Dr. Pankov’s compact embedding theorem when a coercive potential function is used in the energy functional. It should also be noted that Dr. Zhang and Dr. Pankov’s deeply insightful Palais-Smale and linking arguments are used to inspire the bulk of our original linking proof

Materials science of blood clot formation in insects: experimental methods for a multiscale in-situ investigation

This dissertation is structured around the development of a micro and nano-rheological instrument capable of measuring mPa·s-level viscosity of nanoliter droplets and micron thick films in a 10-20 second timeframe and using it to study the kinetics of formation of a blood clot in insects. To understand the materials science behind this clot formation, we enrich the microrheological study with studies of extensional rheology of various maturity stages of clots as well as studies of the surface tension isotherms, dynamic surface tensions, and surface rheology. To study the rapidly changing structure of the clots, we employ high magnification microscopy and scanning electron microscopy. Overall, we perform a detailed study of physical materials properties and structure of the material, which helps us better understand its outstanding performance. In Chapter 1, we introduce an engineering reader to the biological aspect of the problem and discuss the functionality of the material in an insect body. In Chapter 2, we discuss the importance of understanding multiscale rheology of the material and review the current methodologies available and their limitations with regards to the study of changing insect blood. In chapter 3, we discuss the principle of our methodology and the realization of the device with which we study the nanorheology with high precision and temporal resolution. In chapter 4, we present nanoscale viscosity measurements of blood of adult butterflies and moths: Manduca sexta, Vanessa cardui, and Danaus plexippus and discuss the significant deviations of the viscosities from the viscosity of water. In chapter 5, we present the nanorheological measurements of forming and maturing clots in the blood of M. sexta caterpillars and present the discovery of characteristic times of formation of these clots. In chapter 6, we present and discuss the fibrous and cellular structures of the forming blood clots of M. sexta caterpillars. In chapter 7, we study extensional rheology of forming blood clots of M. sexta caterpillars. In chapter 8, we discuss the structure formed in the clots in response to our extensional experiments and relate that to the functions of the clot constituents. Finally, in chapter 9, we study the materials properties of the surface of hemolymph of adult M. sexta, V. cardui, D. plexippus, and caterpillar M. sexta and relate them to the nano and microrheological measurements we performed on the material. We thus characterize the time-dependent structure-properties-performance triangles of blood and the forming blood clots in the studied insects