Understanding the mechanisms behind the emergence and perseverance of cooperation in complex systems is a problem of interest in varied disciplines including biology, physics, economics, the social sciences, and mathematics. The prisoner's dilemma in the setting of evolutionary game theory has become an important framework in which to study this cooperation phenomena. In the past several years, it has been well-documented that models of 'realistic' populations are very favorable to the sustainability of cooperation. In this talk, we will discuss the effect certain population structures (modeled by a graph) have on the success of cooperation in the population. A surprisingly simple relationship between cooperation levels in a population and a graph coefficient which relates the 'connectedness' of an average person to the 'connectedness' of the average neighbor will also be given. This connection will give us a new insight into the 'local' mechanisms which aid cooperation. A natural link to models of the spread of a disease in a population is also provided through this connection.