Let
be a finite group.
Define the graph
on the
set
by putting an edge
between distinct elements
if and only if
is
cyclic. The graph
has appeared in the literature under the names
cyclic graph and
deleted enhanced power graph.
We focus on
for
nontrivial groups
and
. In
particular, when
is connected, we obtain an upper bound on the diameter; in addition, we present an example
meeting this bound. Also, we establish necessary and sufficient conditions for the connectedness
of
.
Finally, we study the connectedness and diameter of
in the
case that
satisfies certain centralizer conditions.
Keywords
cyclic graphs, deleted enhanced power graphs, direct
products