Hamilton Decompositions of Certain 6-regular Cayley Graphs on Abelian Groups with a Cyclic Subgroup of Index Two

Alspach conjectured that every connected Cayley graph of even valency on a finite Abelian group is Hamilton-decomposable. Using some techniques of Liu, this article shows that if A is an Abelian group of even order with a generating set {a,b}, and A contains a subgroup of index two, generated by c, then the 6-regular Cayley graph is Hamilton-decomposable

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

Similar Risk of Kidney Failure among Patients with Blinding Diseases Who Receive Ranibizumab, Aflibercept, and Bevacizumab:An Observational Health Data Sciences and Informatics Network Study

Purpose: To characterize the incidence of kidney failure associated with intravitreal anti-VEGF exposure; and compare the risk of kidney failure in patients treated with ranibizumab, aflibercept, or bevacizumab. Design: Retrospective cohort study across 12 databases in the Observational Health Data Sciences and Informatics (OHDSI) network. Subjects: Subjects aged ≥ 18 years with ≥ 3 monthly intravitreal anti-VEGF medications for a blinding disease (diabetic retinopathy, diabetic macular edema, exudative age-related macular degeneration, or retinal vein occlusion). Methods: The standardized incidence proportions and rates of kidney failure while on treatment with anti-VEGF were calculated. For each comparison (e.g., aflibercept versus ranibizumab), patients from each group were matched 1:1 using propensity scores. Cox proportional hazards models were used to estimate the risk of kidney failure while on treatment. A random effects meta-analysis was performed to combine each database's hazard ratio (HR) estimate into a single network-wide estimate. Main Outcome Measures: Incidence of kidney failure while on anti-VEGF treatment, and time from cohort entry to kidney failure. Results: Of the 6.1 million patients with blinding diseases, 37 189 who received ranibizumab, 39 447 aflibercept, and 163 611 bevacizumab were included; the total treatment exposure time was 161 724 person-years. The average standardized incidence proportion of kidney failure was 678 per 100 000 persons (range, 0–2389), and incidence rate 742 per 100 000 person-years (range, 0–2661). The meta-analysis HR of kidney failure comparing aflibercept with ranibizumab was 1.01 (95% confidence interval [CI], 0.70–1.47; P = 0.45), ranibizumab with bevacizumab 0.95 (95% CI, 0.68–1.32; P = 0.62), and aflibercept with bevacizumab 0.95 (95% CI, 0.65–1.39; P = 0.60). Conclusions: There was no substantially different relative risk of kidney failure between those who received ranibizumab, bevacizumab, or aflibercept. Practicing ophthalmologists and nephrologists should be aware of the risk of kidney failure among patients receiving intravitreal anti-VEGF medications and that there is little empirical evidence to preferentially choose among the specific intravitreal anti-VEGF agents. Financial Disclosures: Proprietary or commercial disclosure may be found in the Footnotes and Disclosures at the end of this article.</p

Small worlds and board interlocking in Brazil: a longitudinal study of corporate networks, 1997-2007

Social Network Analysis (SNA) is an emerging research field in finance, above all in Brazil. This work is pioneering in that it is supported by reference to different areas of knowledge: social network analysis and corporate governance, for dealing with a similarly emerging topic in finance; interlocking boards, the purpose being to check the validity of the small-world model in the Brazilian capital market, and the existence of associations between the positioning of the firm in the network of corporate relationships and its worth. To do so official data relating to more than 400 companies listed in Brazil between 1997 and 2007 were used. The main results obtained suggest that the configuration of the networks of relationships between board members and companies reflects the small-world model. Furthermore, there seems to be a significant relationship between the firm’s centrality and its worth, described according to an “inverted U” curve, which suggests the existence of optimum values of social prominence in the corporate network

Hamilton Decompositions of 6-Regular Cayley Graphs on Even Abelian Groups with Involution-Free Connections Sets

Alspach conjectured that every connected Cayley graph on a finite Abelian group A is Hamilton-decomposable. Liu has shown that for |A| even, if S={s1,…,sk}⊂A is an inverse-free strongly minimal generating set of A, then the Cayley graph Cay(A;S⋆), is decomposable into k Hamilton cycles, whereS⋆ denotes the inverse-closure of S. Extending these techniques and restricting to the 6-regular case, this article relaxes the constraint of strong minimality on S to require only that S be strongly a-minimal, for somea∈S and the index of 〈a〉 be at least four. Strong a-minimality means that 2s∉〈a〉 for all s∈S∖{a,−a}. Some infinite families of open cases for the 6-regular Cayley graphs on even order Abelian groups are resolved. In particular, if |s1|≥|s2|\u3e2|s3|, then Cay(A;{s1,s2,s3}⋆) is Hamilton-decomposable

Matching extendability in Cartesian products of cycles

In a bipartite graph G, a set (Formula presented) is deficient if |N(S)| \u3c |S|. A matching M with vertex set U is k-suitable if G − U has no deficient set of size less than k. Define the extremal function fk (G) to be the largest integer r such that every k-suitable matching in G with at most r edges extends to a perfect matching. Let G(2m)d be the d-fold Cartesian product of the cycle C2m,wherem ≥ 2. We extend results of Vandenbussche and West by showing that for any integers k and d such that (Formula presented), except when m =2 and d =1

n-Isofactorizations of 8-Regular Circulant Graphs

We investigate the problem of decomposing the edges of a connected circulant graph with n vertices and generating set S into isomorphic subgraphs each having n edges. For 8-regular circulants, we show this is always possible when s + 2 ≤ n/4 for all edge lengths s ∈ S

6-regular Cayley graphs on abelian groups of odd order are hamiltonian decomposable

Alspach conjectured that any 2 k-regular connected Cayley graph on a finite abelian group A has a hamiltonian decomposition. In this paper, the conjecture is shown true if k = 3, and the order of A is odd. © 2009 Elsevier B.V. All rights reserved

Ecological Networks over the Edge: Hypergraph Trait-mediated Indirect Interaction (TMII) Structure

Analyses of ecological network structure have yielded important insights into the functioning of complex ecological systems. However, such analyses almost universally omit non-pairwise interactions, many classes of which are crucial for system structure, function, and resilience. Hypergraphs are mathematical constructs capable of considering such interactions: we discuss their utility for studying ecological networks containing diverse interaction types, and associated challenges and strategies. We demonstrate the approach using a real-world coffee agroecosystem in which resistance to agricultural pests depends upon a large number of TMIIs. A hypergraph representation successfully reflects both the importance of species imposing such effects and the context-dependency of that importance in terms of how it is affected by removal of other species from the system