Life is always evolving. The evolutionary dynamics forms the living world around us. At the center of any evolutionary process there exist population of reproduction speci es. These species ca n be molecules, cells, viruses, multi-cellular organisms, strategy in a game or humans with language and different opinions. The evolutionary graph theory is used to study the process of evolution in a network and how the structure of population affects on evolution. This theory is applicable in different fields of research including genetics, sociology, biology, economics, etc. However, because of the complexity of these networks, there are lots of questions about them. One of the important issues that raised in evolutionary dynamics is fixation (extinction) probability (probability that one mutant takeover the whole population). Here, the mutation rate is zero during the evolution. In this case, one of the species becomes extinct during time while other species offspring linage cover the whole population and the fixation in the population. In this thesis, nonzero mutation rate on graphs as well as the networks including two types of species with different fitnesses 1 and r are reviewed by using the Moran process. Each time, one species with a probability proportional to its fitness is selected for reproduction, which replaces a new offspring into one of its neighbors randomly via the mutation probability u. Consequently, the system reachs to a stationary state after enough time. This stationary state is called mutation-selection balance, in which the elimination rate and the reproduction rate of mutations are equal. There are no absorbing states (fixation or extinction) in such system, and therefore both species are present in the system. In this work, the mean number of species is calculated analytically via master equation for the complete and star graphs in stationary state. As a result, this approach ensures that though the use of analytical solutions for the rest of graphs is complicated but the simulation process is applicable for other types of graphs.
