Nordhaus-Gaddum and other bounds for the sum of squares of the positive eigenvalues of a graph

TL;DR

This paper establishes new bounds for the sum of the square roots of the positive eigenvalues' squares of a graph and its complement, extending Nordhaus-Gaddum type inequalities.

Contribution

It proves a new inequality relating the sum of square roots of positive eigenvalues for a graph and its complement, and conjectures a tighter bound.

Findings

Proved that rac{s^{+}(G)}{} + rac{s^{+}(ar{G})}{} < rac{2}{}n.

Used computational tools to search for counterexamples to the conjecture.

Explored bounds for the Randic index in addition to spectral bounds.

Abstract

Terpai [22] proved the Nordhaus-Gaddum bound that $\mu(G) + \mu(\overline{G}) \le 4n/3 - 1$, where $\mu(G)$ is the spectral radius of a graph $G$ with $n$ vertices. Let $s^+$ denote the sum of the squares of the positive eigenvalues of $G$. We prove that $\sqrt{s^{+}(G)} + \sqrt{s^+(\overline{G})} < \sqrt{2}n$ and conjecture that $\sqrt{s^{+}(G)} + \sqrt{s^+(\overline{G})} \le 4n/3 - 1.$ We have used AutoGraphiX and Wolfram Mathematica to search for a counter-example. We also consider Nordhaus-Gaddum bounds for $s^+$ and bounds for the Randi\'c index.