Extremal skew energy of digraphs with no even cycles

Abstract

Let $D$ be a digraph with skew-adjacency matrix $S(D)$. Then the skew energy of $D$ is defined to be the sum of the norms of all eigenvalues of $S(D)$. Denote by $mathcal{O}_n$ the class of digraphs on 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 $n-1leq mleq frac{3}{2}(n-1)$. Then we determine the maximal skew energy digraphs in $mathcal{O}_{n,n}$ and $mathcal{O}_{n,n+1}$, in the latter case assuming that $n$ is even.

Keywords

• skew energy
• digraph
• characteristic polynomial
• signless matching polynomial