205 research outputs found

    Flows and bisections in cubic graphs

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    A kk-weak bisection of a cubic graph GG is a partition of the vertex-set of GG into two parts V1V_1 and V2V_2 of equal size, such that each connected component of the subgraph of GG induced by ViV_i (i=1,2i=1,2) is a tree of at most k−2k-2 vertices. This notion can be viewed as a relaxed version of nowhere-zero flows, as it directly follows from old results of Jaeger that every cubic graph GG with a circular nowhere-zero rr-flow has a ⌊r⌋\lfloor r \rfloor-weak bisection. In this paper we study problems related to the existence of kk-weak bisections. We believe that every cubic graph which has a perfect matching, other than the Petersen graph, admits a 4-weak bisection and we present a family of cubic graphs with no perfect matching which do not admit such a bisection. The main result of this article is that every cubic graph admits a 5-weak bisection. When restricted to bridgeless graphs, that result would be a consequence of the assertion of the 5-flow Conjecture and as such it can be considered a (very small) step toward proving that assertion. However, the harder part of our proof focuses on graphs which do contain bridges.Comment: 14 pages, 6 figures - revised versio

    Internal Partitions of Regular Graphs

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    An internal partition of an nn-vertex graph G=(V,E)G=(V,E) is a partition of VV such that every vertex has at least as many neighbors in its own part as in the other part. It has been conjectured that every dd-regular graph with n>N(d)n>N(d) vertices has an internal partition. Here we prove this for d=6d=6. The case d=n−4d=n-4 is of particular interest and leads to interesting new open problems on cubic graphs. We also provide new lower bounds on N(d)N(d) and find new families of graphs with no internal partitions. Weighted versions of these problems are considered as well

    Graph bisection algorithms

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1986.MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING.Bibliography: leaves 64-66.by Thang Nguyen Bui.Ph.D

    On the minimum bisection of random 3−3-regular graphs

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    In this paper we give new asymptotically almost sure lower and upper bounds on the bisection width of random 3−3-regular graphs. The main contribution is a new lower bound on the bisection width of 0.103295n0.103295n, based on a first moment method together with a structural decomposition of the graph, thereby improving a 27 year old result of Kostochka and Melnikov. We also give a complementary upper bound of 0.139822n0.139822n, combining known spectral ideas with original combinatorial insights. Developping further this approach, with the help of Monte Carlo simulations, we obtain a non-rigorous upper bound of 0.131366n0.131366n.Comment: 48 pages, 20 figure

    Cartan subalgebras in C*-algebras of Hausdorff etale groupoids

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    The reduced C∗C^*-algebra of the interior of the isotropy in any Hausdorff \'etale groupoid GG embeds as a C∗C^*-subalgebra MM of the reduced C∗C^*-algebra of GG. We prove that the set of pure states of MM with unique extension is dense, and deduce that any representation of the reduced C∗C^*-algebra of GG that is injective on MM is faithful. We prove that there is a conditional expectation from the reduced C∗C^*-algebra of GG onto MM if and only if the interior of the isotropy in GG is closed. Using this, we prove that when the interior of the isotropy is abelian and closed, MM is a Cartan subalgebra. We prove that for a large class of groupoids GG with abelian isotropy---including all Deaconu--Renault groupoids associated to discrete abelian groups---MM is a maximal abelian subalgebra. In the specific case of kk-graph groupoids, we deduce that MM is always maximal abelian, but show by example that it is not always Cartan.Comment: 14 pages. v2: Theorem 3.1 in v1 incorrect (thanks to A. Kumjain for pointing out the error); v2 shows there is a conditional expectation onto MM iff the interior of the isotropy is closed. v3: Material (including some theorem statements) rearranged and shortened. Lemma~3.5 of v2 removed. This version published in Integral Equations and Operator Theor

    Towards Resistance Sparsifiers

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    We study resistance sparsification of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular expander admits a (1+ϵ)(1+\epsilon)-resistance sparsifier of size O~(n/ϵ)\tilde O(n/\epsilon), and conjecture this bound holds for all graphs on nn nodes. In comparison, spectral sparsification is a strictly stronger notion and requires Ω(n/ϵ2)\Omega(n/\epsilon^2) edges even on the complete graph. Our approach leads to the following structural question on graphs: Does every dense regular expander contain a sparse regular expander as a subgraph? Our main technical contribution, which may of independent interest, is a positive answer to this question in a certain setting of parameters. Combining this with a recent result of von Luxburg, Radl, and Hein~(JMLR, 2014) leads to the aforementioned resistance sparsifiers

    Deleterious effect of suboptimal diet on rest-activity cycle in Anastrepha ludens manifests itself with age.

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    Activity patterns and sleep-wake cycles are among the physiological processes that change most prominently as animals age, and are often good indicators of healthspan. In this study, we used the video-based high-resolution behavioral monitoring system (BMS) to monitor the daily activity cycle of tephritid fruit flies Anastrepha ludens over their lifetime. Surprisingly, there was no dramatic change in activity profile with respect to age if flies were consistently fed with a nutritionally balanced diet. However, if flies were fed with sugar-only diet, their activity profile decreased in amplitude at old age, suggesting that suboptimal diet affected activity patterns, and its detrimental effect may not manifest itself until the animal ages. Moreover, by simulating different modes of behavior monitoring with a range of resolution and comparing the resulting conclusions, we confirmed the superior performance of video-based monitoring using high-resolution BMS in accurately representing activity patterns in an insect model
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