  On a finite graph with a chosen partition of the vertex set into interior and
boundary vertices, a $\lambda$-polyharmonic function is a complex function $f$
on the vertex set which satisfies $(\lambda \cdot I - P)^n f(x) = 0$ at each
interior vertex. Here, $P$ may be the normalised adjaceny matrix, but more
generally, we consider the transition matrix $P$ of an arbitrary Markov chain
to which the (oriented) graph structure is adapted. After describing these
`global' polyharmonic functions, we turn to solving the Riquier problem, where
$n$ boundary functions are preassigned and a corresponding `tower' of $n$
successive Dirichlet type problems are solved. The resulting unique solution
will be polyharmonic only at those points which have distance at least $n$ from
the boundary. Finally, we compare these results with those concerning infinite
trees with the end boundary, as studied by Cohen, Colonnna, Gowrisankaran and
Singman, and more recently, by Picardello and Woess.
