  We study functional graphs generated by quadratic polynomials over prime
fields. We introduce efficient algorithms for methodical computations and
provide the values of various direct and cumulative statistical parameters of
interest. These include: the number of connected functional graphs, the number
of graphs having a maximal cycle, the number of cycles of fixed size, the
number of components of fixed size, as well as the shape of trees extracted
from functional graphs. We particularly focus on connected functional graphs,
that is, the graphs which contain only one component (and thus only one cycle).
Based on the results of our computations, we formulate several conjectures
highlighting the similarities and differences between these functional graphs
and random mappings.
