  This paper considers the degree-diameter problem for undirected circulant
graphs. The focus is on extremal graphs of given (small) degree and arbitrary
diameter. The published literature only covers graphs of up to degree 7. The
approach used to establish the results for degree 6 and 7 has been extended
successfully to degree 8 and 9. Candidate graphs are defined as functions of
the diameter for both degree 8 and degree 9. They are proven to be extremal for
small diameters. They establish new lower bounds for all greater diameters, and
are conjectured to be extremal. The existence of the degree 8 solution is
proved for all diameters. Finally some conjectures are made about solutions for
circulant graphs of higher degree.
