  One of the most fundamental concepts of evolutionary dynamics is the
"fixation" probability, i.e. the probability that a mutant spreads through the
whole population. Most natural communities are geographically structured into
habitats exchanging individuals among each other and can be modeled by an
evolutionary graph (EG), where directed links weight the probability for the
offspring of one individual to replace another individual in the community.
Very few exact analytical results are known for EGs. We show here how by using
the techniques of the fixed point of Probability Generating Function, we can
uncover a large class of of graphs, which we term bithermal, for which the
exact fixation probability can be simply computed.
