  We explore graph theoretical properties of minimal prime graphs of finite
solvable groups. In finite group theory studying the prime graph of a group has
been an important topic for the past almost half century. Recently prime graphs
of solvable groups have been characterized in graph theoretical terms only.
This now allows the study of these graphs with methods from graph theory only.
Minimal prime graphs turn out to be of particular interest, and in this paper
we pursue this further by exploring, among other things, diameters, Hamiltonian
cycles and the property of being self-complementary for minimal prime graphs.
We also study a new, but closely related notion of minimality for prime graphs
and look into counting minimal prime graphs.
