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

Terpai [22] proved the Nordhaus-Gaddum bound that $\mu(G) + \mu(\overline{G}) \le 4n/3 - 1$,where$\mu(G)$isthespectralradiusofagraph$G$with$n$vertices\mathrm{.}Let$s^+$ denote the sum of the squares of the positive eigenvalues of $G$. We prove that $\sqrt{s^{+}(G)}+\sqrt{s^{+}(\bar{G})}<\sqrt{2}n$ and conjecture that $\sqrt{s^{+}(G)}+\sqrt{s^{+}(\bar{G})}\le 4n\mathrm{/}3-1.$ 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.