In this paper, we characterize the extremal digraphs with the maximal or minimal $\alpha$-spectral radius among some digraph classes such as rose digraphs, generalized theta digraphs and tri-ring digraphs with given size $m$. These digraph classes are denoted by $\mathcal{ R}_{ m}^k$, $\widetilde{ \boldsymbol{ \Theta}}_k(m)$ and $\INF(m)$ respectively. The main results about spectral extremal digraph by Guo and Liu in \cite{ MR2954483} and Li and Wang in \cite{ MR3777498} are generalized to $\alpha$-spectral graph theory. As a by-product of our main results, an open problem in \cite{ MR3777498} is answered. Furthermore, we determine the digraphs with the first three minimal $\alpha$-spectral radius among all strongly connected digraphs. Meanwhile, we determine the unique digraph with the fourth minimal $\alpha$-spectral radius among all strongly connected digraphs for $0\le \alpha \le \frac{ 1}{ 2}$.
Categories: math.CO