Properties of random coverings of graphs

Wydział Matematyki i InformatykiPrzedmiotem rozprawy doktorskiej są asymptotyczne własności losowych nakryć grafów zdefiniowanych przez Amita i Liniala w 2002 roku, jako nowy model grafu losowego. Dla zadanego grafu bazowego G losowe nakrycie stopnia n, oznaczane jako Ḡ, otrzymujemy poprzez zastąpienie każdego wierzchołka v przez n-elementowy zbiór Ḡ_v oraz wybór, dla każdej krawędzi e={x,y} \in E(G), z jednostajnym prawdopodobieństwem, losowego skojarzenia pomiędzy zbiorami Ḡ_x i Ḡ_y. Pierwszym zagadnieniem poruszanym w pracy jest oszacowanie wielkości największej topologicznej kliki zawartej (jako podgraf) w losowym nakryciu danego grafu G. Udało się pokazać, że asymptotycznie prawie na pewno losowe nakrycie grafu G zawiera największą z możliwych topologiczną klikę. Drugim badanym zagadnieniem jest pytanie o istnienie w podniesieniu grafu cyklu Hamiltona. W pracy pokazujemy, że jeżeli graf G zawiera dwa krawędziowo rozłączne cykle Hamiltona, których suma nie jest grafem dwudzielnym i ma minimalny stopień co najmniej 5, to asymptotycznie prawie na pewno nakrycie Ḡ jest grafem hamiltonowskim.In the thesis we study selected properties of random coverings of graphs introduced by Amit and Linial in 2002. A random n-covering of a graph G, denoted by Ḡ, is obtained by replacing each vertex v of G by an n-element set Ḡ_v and then choosing, independently for every edge e = {x,y}\in E(G), uniformly at random a perfect matching between Ḡ_x and Ḡ_y. The first problem we consider is the typical size of the largest topological clique in random covering of given graph G. We show that asymptotically almost surely a random n-covering of a graph G contains the largest possible topological clique. The second property we examine is the existence of a Hamilton cycle in Ḡ. We show that if G contains two edge disjoint Hamilton cycles, which sum is not a bipartite graph and has minimum degree at least 5, then asymptotically almost surely Ḡ is Hamiltonian

European Journal of Combinatorics Index, Volume 27

BACKGROUND: Diabetes is an inflammatory condition associated with iron abnormalities and increased oxidative damage. We aimed to investigate how diabetes affects the interrelationships between these pathogenic mechanisms. METHODS: Glycaemic control, serum iron, proteins involved in iron homeostasis, global antioxidant capacity and levels of antioxidants and peroxidation products were measured in 39 type 1 and 67 type 2 diabetic patients and 100 control subjects. RESULTS: Although serum iron was lower in diabetes, serum ferritin was elevated in type 2 diabetes (p = 0.02). This increase was not related to inflammation (C-reactive protein) but inversely correlated with soluble transferrin receptors (r = - 0.38, p = 0.002). Haptoglobin was higher in both type 1 and type 2 diabetes (p &lt; 0.001) and haemopexin was higher in type 2 diabetes (p &lt; 0.001). The relation between C-reactive protein and haemopexin was lost in type 2 diabetes (r = 0.15, p = 0.27 vs r = 0.63, p &lt; 0.001 in type 1 diabetes and r = 0.36, p = 0.001 in controls). Haemopexin levels were independently determined by triacylglycerol (R(2) = 0.43) and the diabetic state (R(2) = 0.13). Regarding oxidative stress status, lower antioxidant concentrations were found for retinol and uric acid in type 1 diabetes, alpha-tocopherol and ascorbate in type 2 diabetes and protein thiols in both types. These decreases were partially explained by metabolic-, inflammatory- and iron alterations. An additional independent effect of the diabetic state on the oxidative stress status could be identified (R(2) = 0.5-0.14). CONCLUSIONS: Circulating proteins, body iron stores, inflammation, oxidative stress and their interrelationships are abnormal in patients with diabetes and differ between type 1 and type 2 diabetes</p

Hamilton decompositions of 6-regular abelian Cayley graphs

In 1969, Lovasz asked whether every connected, vertex-transitive graph has a Hamilton path. This question has generated a considerable amount of interest, yet remains vastly open. To date, there exist no known connected, vertex-transitive graph that does not possess a Hamilton path. For the Cayley graphs, a subclass of vertex-transitive graphs, the following conjecture was made: Weak Lovász Conjecture: Every nontrivial, finite, connected Cayley graph is hamiltonian. The Chen-Quimpo Theorem proves that Cayley graphs on abelian groups flourish with Hamilton cycles, thus prompting Alspach to make the following conjecture: Alspach Conjecture: Every 2k-regular, connected Cayley graph on a finite abelian group has a Hamilton decomposition. Alspach’s conjecture is true for k = 1 and 2, but even the case k = 3 is still open. It is this case that this thesis addresses. Chapters 1–3 give introductory material and past work on the conjecture. Chapter 3 investigates the relationship between 6-regular Cayley graphs and associated quotient graphs. A proof of Alspach’s conjecture is given for the odd order case when k = 3. Chapter 4 provides a proof of the conjecture for even order graphs with 3-element connection sets that have an element generating a subgroup of index 2, and having a linear dependency among the other generators. Chapter 5 shows that if Γ = Cay(A, {s1, s2, s3}) is a connected, 6-regular, abelian Cayley graph of even order, and for some1 ≤ i ≤ 3, Δi = Cay(A/(si), {sj1 , sj2}) is 4-regular, and Δi ≄ Cay(ℤ3, {1, 1}), then Γ has a Hamilton decomposition. Alternatively stated, if Γ = Cay(A, S) is a connected, 6-regular, abelian Cayley graph of even order, then Γ has a Hamilton decomposition if S has no involutions, and for some s ∈ S, Cay(A/(s), S) is 4-regular, and of order at least 4. Finally, the Appendices give computational data resulting from C and MAGMA programs used to generate Hamilton decompositions of certain non-isomorphic Cayley graphs on low order abelian groups

Combinatorics

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Unsolved Problems in Spectral Graph Theory

Spectral graph theory is a captivating area of graph theory that employs the eigenvalues and eigenvectors of matrices associated with graphs to study them. In this paper, we present a collection of $20$ topics in spectral graph theory, covering a range of open problems and conjectures. Our focus is primarily on the adjacency matrix of graphs, and for each topic, we provide a brief historical overview.Comment: v3, 30 pages, 1 figure, include comments from Clive Elphick, Xiaofeng Gu, William Linz, and Dragan Stevanovi\'c, respectively. Thanks! This paper will be published in Operations Research Transaction

A Survey on Graph Kernels

Graph kernels have become an established and widely-used technique for solving classification tasks on graphs. This survey gives a comprehensive overview of techniques for kernel-based graph classification developed in the past 15 years. We describe and categorize graph kernels based on properties inherent to their design, such as the nature of their extracted graph features, their method of computation and their applicability to problems in practice. In an extensive experimental evaluation, we study the classification accuracy of a large suite of graph kernels on established benchmarks as well as new datasets. We compare the performance of popular kernels with several baseline methods and study the effect of applying a Gaussian RBF kernel to the metric induced by a graph kernel. In doing so, we find that simple baselines become competitive after this transformation on some datasets. Moreover, we study the extent to which existing graph kernels agree in their predictions (and prediction errors) and obtain a data-driven categorization of kernels as result. Finally, based on our experimental results, we derive a practitioner's guide to kernel-based graph classification