3,933 research outputs found

### 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 $k$-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 $E$-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

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