  A Frobenius group is a transitive but not regular permutation group such that
only the identity element can fix two points. A finite Frobenius group can be
expressed as $G = K \rtimes H$ with $K$ a nilpotent normal subgroup. A
first-kind $G$-Frobenius graph is a Cayley graph on $K$ with connection set $S$
an $H$-orbit on $K$ generating $K$, where $H$ is of even order or $S$ consists
of involutions. We classify all 6-valent first-kind Frobenius circulant graphs
such that the underlying kernel $K$ is cyclic. We give optimal gossiping and
routing algorithms for such a circulant and compute its forwarding indices,
Wiener indices and minimum gossip time. We also prove that its broadcasting
time is equal to its diameter plus two or three. We prove that all 6-valent
first-kind Frobenius circulants with cyclic kernels are Eisenstein-Jacobi
graphs, the latter being Cayley graphs on quotient rings of the ring of
Eisenstein-Jacobi integers. We also prove that larger Eisenstein-Jacobi graphs
can be constructed from smaller ones as topological covers, and a similar
result holds for 6-valent first-kind Frobenius circulants. As a corollary any
Eisenstein-Jacobi graph with order congruent to 1 modulo 6 and underlying
Eisenstein-Jacobi integer not an associate of a real integer, is a cover of a
6-valent first-kind Frobenius circulant. A distributed real-time computing
architecture known as HARTS or hexagonal mesh is a special 6-valent first-kind
Frobenius circulant.
