Nordhaus-Gaddum inequality for the spectral radius of a graph of order $n$

TL;DR

This paper identifies the extremal graph that maximizes the sum of spectral radii of a graph and its complement, resolving a conjecture from 2007.

Contribution

It provides a definitive solution to the Nordhaus-Gaddum inequality for spectral radius, confirming a longstanding conjecture.

Findings

The extremal graph configuration is characterized.

The conjecture by Stevanović is proven true.

The result advances understanding of spectral graph theory.

Abstract

We determine the extremal graph $G$ of order $n$ that maximizes the sum of the spectral radii of $G$ and its complement. This resolves a conjecture posed by Stevanovi\'{c} in 2007.