  Population structure affects the outcome of natural selection. Static
population structures can be described by graphs, where individuals occupy the
nodes, and interactions occur along the edges. General conditions for
evolutionary success on any weighted graph were recently derived, for weak
selection, in terms of coalescence times of random walks. Here we show that for
a special class of graphs, the conditions for success take a particularly
simple form, in which all effects of graph structure are described by the
graph's "effective degree"---a measure of the effective number of neighbors per
individual. This result holds for all weighted graphs that are isothermal,
meaning that the sum of edge weights is the same at each node. Isothermal
graphs encompass a wide variety of underlying topologies, and arise naturally
from supposing that each individual devotes the same amount of time to
interaction. Cooperative behavior is favored on a large isothermal graph if the
benefit-to-cost ratio exceeds the effective degree. We relate the effective
degree of a graph to its spectral gap, thereby providing a link between
evolutionary dynamics and the theory of expander graphs. As a surprising
example, we report graphs of infinite average degree that are nonetheless
highly conducive for promoting cooperation.
