The original motivation for finding expander families was to build economical robust networks for telephone and computer communication. Over the past three decades, expander families have been developed into a powerful tool with wide applications in many areas such as fast distributed routing algorithms, storage schemes, telecommunication, and cryptography. This thesis focuses on the theoretical aspects of expander families. The primary goal is to apply spectral graph theory to show the non-existence of an expander family within the class of circulant graphs. There are other proofs out there, but this is a fundamental approach. Another part of this thesis uses the adjacency matrix and its properties to prove Cheeger's inequalities and determine when the equalities hold.