  We study the emergence of cooperation in structured populations with any
arrangement of cooperators and defectors on the evolutionary graph. Using
structure coefficients defined for configurations describing such arrangements
of any number of mutants, we provide results for weak selection to favor
cooperation over defection on any regular graph with $N \leq 14$ vertices.
Furthermore, the properties of graphs that particularly promote cooperation are
analyzed. It is shown that the number of graph cycles of certain length is a
good predictor for the values of the structure coefficient, and thus a tendency
to favor cooperation. Another property of particularly cooperation-promoting
regular graphs with a low degree is that they are structured to have blocks
with clusters of mutants that are connected by cut vertices and/or hinge
vertices.
