  We use production matrices to count several classes of geometric graphs. We
present novel production matrices for non-crossing partitions, connected
geometric graphs, and k-angulations, which provide another way of counting the
number of such objects. Counting geometric graphs is then equivalent to
calculating the powers of a production matrix. Applying the technique of
Riordan Arrays to these production matrices, we establish new formulas for the
numbers of geometric graphs as well as combinatorial identities derived from
the production matrices. Further, we obtain the characteristic polynomial and
the eigenvectors of such production matrices.
