Constructions of strongly regular Cayley graphs using index four Gauss sums

We give a construction of strongly regular Cayley graphs on finite fields \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}_{q}$\end{document} by using union of cyclotomic classes and index 4 Gauss sums. In particular, we obtain two infinite families of strongly regular graphs with new parameters.