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In this paper, we give constructions of strongly regular Cayley graphs andskew Hadamard difference sets. Both constructions are based on choosingcyclotomic classes in finite fields, and our results generalize ten of theeleven sporadic examples of cyclotomic strongly regular graphs given by Schmidtand White [24] and several of subfield examples into infinite families. Theseinfinite families of strongly regular graphs have new parameters. The maintools that we employed are relative Gauss sums instead of explicit evaluationsof Gauss sums.

Strongly Regular Cayley Graphs, Skew Hadamard Difference Sets, and Rationality of Relative Gauss Sums