Abstract
Let G be a finite group and V(G) be the set of all non-central conjugacy classes of G. Consider the conjugacy class graph of G which is defined as follows: its vertex set is the set V(G) and two distinct vertices xG and yG are connected with an edge if o(x) and o(y) are not coprime. In this article, we determine the structure of groups G whose non-trivial elements have prime orders and whose graph does not have any complete subgraph with six vertices.
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