  We study a general transition operator, generated by a random walk on a graph
$X$; in particular we give necessary and sufficient condition on the matrix
coefficient (1-step transition probablilities) to be a bounded operator from
$l^\infty(X)$ into itself. Moreover we characterize compact operators and we
relate this property to the behaviour of the associated random walk. We give a
necessary and sufficient condition for the pre-adjoint of the discrete Laplace
operator to be an injective map.
