1Department of Applied Mathematics, Northwestern Polytechnical University
2Center for Combinatorics and LPMC-TJKLC, Nankai University
3Center for Combinatorics, Nankai University
Abstract
Let $D$ be a digraph with skew-adjacency matrix $S(D)$. Then the skew energy of $D$ is defined as the sum of the norms of all eigenvalues of $S(D)$. Denote by $\mathcal{O}_n$ the class of digraphs of order $n$ with no even cycles, and by $\mathcal{O}_{n,m}$ the class of digraphs in $\mathcal{O}_n$ with $m$ arcs. In this paper, we first give the minimal skew energy digraphs in $\mathcal{O}_n$ and $\mathcal{O}_{n,m}$ with $n-1\leq m\leq \frac{3}{2}(n-1)$. Then we determine the maximal skew energy digraphs in $\mathcal{O}_{n,n}$ and $\mathcal{O}_{n,n+1}$, and in the latter case we assume that $n$ is even.
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