  We analyze site percolation on directed and undirected graphs with
site-dependent open-site probabilities. We construct upper bounds on cluster
susceptibilities, vertex connectivity functions, and the expected number of
simple open cycles through a chosen arc; separate bounds are given on finite
and infinite (di)graphs. These produce lower bounds for percolation and
uniqueness transitions in infinite (di)graphs, and for the formation of a giant
component in finite (di)graphs. The bounds are formulated in terms of
appropriately weighted adjacency and non-backtracking (Hashimoto) matrices. It
turns out to be the uniqueness criterion that is most closely associated with
an asymptotically vanishing probability of forming a giant strongly-connected
component on a large finite (di)graph.
