  Let C be an integral fusion category. We study some graphs, called the prime
graph and the common divisor graph, related to the Frobenius-Perron dimensions
of simple objects in the category C, that extend the corresponding graphs
associated to the irreducible character degrees and the conjugacy class sizes
of a finite group. We describe these graphs in several cases, among others,
when C is an equivariantization under the action of a finite group, a 2-step
nilpotent fusion category, and the representation category of a twisted quantum
double. We prove generalizations of known results on the number of connected
components of the corresponding graphs for finite groups in the context of
braided fusion categories. In particular, we show that if C is any integral
non-degenerate braided fusion category, then the prime graph of C has at most 3
connected components, and it has at most 2 connected components if C is in
addition solvable. As an application we prove a classification result for
weakly integral braided fusion categories all of whose simple objects have
prime power Frobenius-Perron dimension.
