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

Terpai [21] 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‾)<2n and 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.