3,723 research outputs found

    The Maximum Likelihood Threshold of a Graph

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    The maximum likelihood threshold of a graph is the smallest number of data points that guarantees that maximum likelihood estimates exist almost surely in the Gaussian graphical model associated to the graph. We show that this graph parameter is connected to the theory of combinatorial rigidity. In particular, if the edge set of a graph GG is an independent set in the n1n-1-dimensional generic rigidity matroid, then the maximum likelihood threshold of GG is less than or equal to nn. This connection allows us to prove many results about the maximum likelihood threshold.Comment: Added Section 6 and Section

    Sandwiching saturation number of fullerene graphs

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    The saturation number of a graph GG is the cardinality of any smallest maximal matching of GG, and it is denoted by s(G)s(G). Fullerene graphs are cubic planar graphs with exactly twelve 5-faces; all the other faces are hexagons. They are used to capture the structure of carbon molecules. Here we show that the saturation number of fullerenes on nn vertices is essentially n/3n/3

    Triangle-Free Penny Graphs: Degeneracy, Choosability, and Edge Count

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    We show that triangle-free penny graphs have degeneracy at most two, list coloring number (choosability) at most three, diameter D=Ω(n)D=\Omega(\sqrt n), and at most min(2nΩ(n),2nD2)\min\bigl(2n-\Omega(\sqrt n),2n-D-2\bigr) edges.Comment: 10 pages, 2 figures. To appear at the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017
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