Some $\alpha$-spectral extremal results for some digraphs

TL;DR

This paper characterizes extremal digraphs with maximal or minimal $ ext{alpha}$-spectral radius within specific classes, generalizing previous spectral extremal results and solving an open problem in the field.

Contribution

It extends spectral extremal results to $ ext{alpha}$-spectral graph theory for various digraph classes and identifies extremal digraphs with minimal and maximal spectral radii.

Findings

Identified extremal digraphs with maximal/minimal $ ext{alpha}$-spectral radius in certain classes.

Solved an open problem related to $ ext{alpha}$-spectral extremal digraphs.

Determined the first three and the fourth minimal $ ext{alpha}$-spectral radius digraphs among strongly connected digraphs.

Abstract

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 $0\le \alpha \le \frac{1}{2}$.