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Let \(\Gamma\) be a finite group and let \(X\) be a subset of \(\Gamma\) such that \(X^{-1} = X\) and \(1 \notin X\). The conjugacy graph \(\text{Con}(\Gamma; X)\) has vertex set \(\Gamma\) and two vertices \(g, h \in \Gamma\) are adjacent in \(\text{Con}(\Gamma; X)\) if and only if there exists \(x \in X\) with \(g = xhx^{-1}\). The components of a conjugacy graph partition the vertices into conjugacy classes (with respect to \(X\)) of the group. Sufficient conditions for a conjugacy graph to have either vertex-transitive or arc-transitive components are provided. It is also shown that every Cayley graph is the component of some conjugacy graph.