On the spectral radius of simple digraphs with prescribed number of arcs

TL;DR

This paper establishes a precise upper bound for the spectral radius of simple digraphs with a fixed number of arcs and characterizes the extremal graphs that achieve this maximum.

Contribution

It provides a sharp upper bound for the spectral radius and characterizes all extremal simple digraphs with maximum spectral radius for specific arc counts.

Findings

Derived a sharp upper bound for spectral radius.

Characterized extremal simple digraphs with maximum spectral radius.

Identified conditions for extremal graphs with given arc counts.

Abstract

This paper presents a sharp upper bound for the spectral radius of simple digraphs with described number of arcs. Further, the extremal graphs which attain the maximum spectral radius among all simple digraphs with fixed arcs are investigated. In particular, we characterize all extremal simple digraphs with the maximum spectral radius among all simple digraphs with arcs number $e=2{k\choose 2}+t$ and $k>4t^4+4$.