  A Lie algebra is Frobenius if it admits a linear functional F such that the
Kirillov form F([x,y]) is non-degenerate. If g is the m-th maximal parabolic
subalgebra P(n,m) of sl(n) this occurs precisely when (n,m) = 1. We define a
"cyclic" functional F on P(n,m) and prove it is non-degenerate using properties
of certain graphs associated to F. These graphs also provide in some cases
readily computable associated solutions of the classical Yang-Baxter equation.
We also define a local ring associated to each connected loopless graph from
which we show that the graph can be reconstructed. Finally, we examine the
seaweed Lie algebras of Dergachev and Kirillov from our perspective.
