Some α-spectral extremal results for some digraphs.

In this paper, we characterize the extremal digraphs with the maximal or minimal α-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 R_m^k, ...(m) and ...(m) respectively. The main results about spectral extremal digraph by Guo and Liu [Some results on the spectral radius of generalized ∞ and θ-digraphs. Linear Algebra Appl. 2012;437(9):2200-2208] and Li et al. [The signless Laplacian spectral radius of some strongly connected digraphs. Indian J Pure Appl Math. 2018;49(1):113-127] are generalized to α-spectral graph theory. As a by-product of our main results, an open problem in Li et al. [The signless Laplacian spectral radius of some strongly connected digraphs. Indian J Pure Appl Math. 2018;49(1):113-127] is answered. Furthermore, we determine the digraphs with the first three minimal α-spectral radius among all strongly connected digraphs. Meanwhile, we determine the unique digraph with the fourth minimal α-spectral radius among all strongly connected digraphs for 0 ≤ α ≤ 1/2. [ABSTRACT FROM AUTHOR]