Some α-spectral extremal results for some digraphs

Abstract

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 ${\mathcal{R}}_m^k$, ${\tilde{\mathbf{\Theta }}}_k\left(m\right)$ 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 $0\le \alpha \le \frac{1}{2}$.

Keywords:

• Digraph
• α-sectral radius
• rose digraphs
• tri-ring digraphs

AMS subject classifications 2010:

• 05C20
• 05C50

Disclosure statement

No potential conflict of interest was reported by the author(s).