Eigenvalue problems of Nordhaus-Gaddum type

TL;DR

This paper extends Nordhaus-Gaddum type bounds from spectral radii to other eigenvalues of graphs, providing new bounds and solving the problem for second eigenvalues.

Contribution

It generalizes existing bounds to a broader set of eigenvalues and offers solutions specifically for the second eigenvalues of graphs.

Findings

Derived bounds for eigenvalues of graphs and their complements.

Solved the problem for the second eigenvalues.

Extended Nordhaus-Gaddum inequalities to new spectral parameters.

Abstract

In 1970 Nosal gave upper and lower bounds on the sum of the spectral radii of a graph and its complement. We generalize this problem to other eigenvalues and give a number of bounds. We essentially solve the corresponding problem for the second eigenvalues.