  We develop a theory of average sizes of kernels of generic matrices with
support constraints defined in terms of graphs and hypergraphs. We apply this
theory to study unipotent groups associated with graphs. In particular, we
establish strong uniformity results pertaining to zeta functions enumerating
conjugacy classes of these groups. We deduce that the numbers of conjugacy
classes of $\mathbf{F}_q$-points of the groups under consideration depend
polynomially on $q$. Our approach combines group theory, graph theory, toric
geometry, and $p$-adic integration.
  Our uniformity results are in line with a conjecture of Higman on the numbers
of conjugacy classes of unitriangular matrix groups. Our findings are, however,
in stark contrast to related results by Belkale and Brosnan on the numbers of
generic symmetric matrices of given rank associated with graphs.
