Extremal skew energy of digraphs with no even cycles

Document Type: Research Paper

Authors

1 Department of Applied Mathematics, Northwestern Polytechnical University

2 Center for Combinatorics and LPMC-TJKLC, Nankai University

3 Center 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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