In this thesis, we apply spectral graph theory to show the non-existence of an
expander family within the class of circulant graphs. Using the adjacency matrix and its properties, we prove Cheeger\u27s inequalities and determine when the equalities hold. In order to apply Cheeger\u27s inequalities, we compute the spectrum of a general circulant graph and approximate its second largest eigenvalue. Finally, we show that circulant graphs do not contain an expander family
