  The rank three subgroups of a one-dimensional affine group over a finite
field were classified in 1978 by Foulser and Kallaher. Although one can use
their results for a classification of corresponding rank three graphs, the
author did not find such a classification in a literature. The goal of this
note is to present such a classification. It turned out that graph
classification is much simpler than the group one. More precisely, it is shown
that the graphs in the title are either the Paley graphs or one of the graphs
constructed by Van Lint and Schrijver or by Peisert. Our approach is based on
elementary group theory and does not use the classification of rank three
affine groups.
