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Let be a finite
-solvable group. Attach to
the following graph
: its vertices are the non-central conjugacy classes of
-regular elements of
, and two vertices are connected by an edge if their cardinalities are not coprime. We prove that the number of connected components of
is at most 2. When
is connected, then the diameter of the graph is at most 3, and when
is disconnected, then each of the two components is a complete graph.
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