The skew energy of random oriented graphs

Given a graph $G$, let $G^\sigma$ be an oriented graph of $G$ with the orientation $\sigma$ and skew-adjacency matrix $S(G^\sigma)$. The skew energy of the oriented graph $G^\sigma$, denoted by $\mathcal{E}_S(G^\sigma)$, is defined as the sum of the absolute values of all the eigenvalues of $S(G^\sigma)$. 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 $G_{n,d}^\sigma$, 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

A path in an edge-colored graph $G$ 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 $G$ such that every pair of vertices are connected by at least $k$ internally vertex-disjoint rainbow paths is called the rainbow $k$-connectivity of the graph $G$, denoted by $rc_k(G)$. For the random graph $G(n,p)$, He and Liang got a sharp threshold function for the property $rc_k(G(n,p))\leq d$. In this paper, we extend this result to the case of random bipartite graph $G(m,n,p)$.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1012.1942 by other author

Picard groups and duality for Real Morava EE-theories

We show, at the prime 2, that the Picard group of invertible modules over $E_n^{hC_2}$ is cyclic. Here, $E_n$ is the height $n$ Lubin--Tate spectrum and its $C_2$-action is induced from the formal inverse of its associated formal group law. We further show that $E_n^{hC_2}$ is Gross--Hopkins self-dual and determine the exact shift. Our results generalize the well-known results when $n = 1$.Comment: Comments welcome. Abstract and introduction update