3,933 research outputs found

    The skew energy of random oriented graphs

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    Given a graph GG, let GσG^\sigma be an oriented graph of GG with the orientation σ\sigma and skew-adjacency matrix S(Gσ)S(G^\sigma). The skew energy of the oriented graph GσG^\sigma, denoted by ES(Gσ)\mathcal{E}_S(G^\sigma), is defined as the sum of the absolute values of all the eigenvalues of S(Gσ)S(G^\sigma). In this paper, we study the skew energy of random oriented graphs and formulate an exact estimate of the skew energy for almost all oriented graphs by generalizing Wigner's semicircle law. Moreover, we consider the skew energy of random regular oriented graphs Gn,dσG_{n,d}^\sigma, and get an exact estimate of the skew energy for almost all regular oriented graphs.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1011.6646 by other author

    Rainbow kk-connectivity of random bipartite graphs

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    A path in an edge-colored graph GG is called a rainbow path if no two edges of the path are colored the same. The minimum number of colors required to color the edges of GG such that every pair of vertices are connected by at least kk internally vertex-disjoint rainbow paths is called the rainbow kk-connectivity of the graph GG, denoted by rck(G)rc_k(G). For the random graph G(n,p)G(n,p), He and Liang got a sharp threshold function for the property rck(G(n,p))≀drc_k(G(n,p))\leq d. In this paper, we extend this result to the case of random bipartite graph G(m,n,p)G(m,n,p).Comment: 15 pages. arXiv admin note: text overlap with arXiv:1012.1942 by other author

    Picard groups and duality for Real Morava EE-theories

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    We show, at the prime 2, that the Picard group of invertible modules over EnhC2E_n^{hC_2} is cyclic. Here, EnE_n is the height nn Lubin--Tate spectrum and its C2C_2-action is induced from the formal inverse of its associated formal group law. We further show that EnhC2E_n^{hC_2} is Gross--Hopkins self-dual and determine the exact shift. Our results generalize the well-known results when n=1n = 1.Comment: Comments welcome. Abstract and introduction update
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