There is a growing interest in the study of evolutionary dynamics on populations with some non-homogeneous structure. In this paper we follow the model of Lieberman et al. [Lieberman, E., Hauert, C., Nowak, M.A. 2005 Evolutionary dynamics on graphs. Nature 43 (3), 312-316] of evolutionary dynamics on a graph. We investigate the case of non-directed equally weighted graphs and find solutions for the fixation probability of a single mutant in two classes of simple graphs. We further demonstrate

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... at finding similar solutions on graphs outside these classes is far more complex. Finally we investigate our chosen classes numerically, and discuss a number of features of the graphs; for example we find the fixation probabilities for different initial starting positions and observe that average fixation probabilities are always increased for advantageous mutants as compared against those of unstructured populations.