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

Abstract

Terpai [22] proved the Nordhaus-Gaddum bound that µ(G) + µ(G) ≤ 4n/3 − 1, where µ(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 s + (G)+ s + (G) < √ 2nand conjecture that s + (G) + s + (G) ≤ 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 Randić index.