Summary: A dissociation set of a graph \(G\) is a set of vertices which induces a subgraph of \(G\) with maximum degree at most 1, or equivalently, a set of vertices whose complement in \(G\) is a 3-path vertex cover (intersecting every 3-path of \(G)\). The maximum cardinality of a dissociation set of \(G\) is called the dissociation number of \(G\). We study the dissociation number of a circulant graph (a Cayley graph of the group \(\mathbb{Z}_n)\) and generalize this concept to the dissociation ratio of an integer distance graph (a Cayley graph of the group \(\mathbb{Z} )\).