In this paper a new graph based representation of Boolean formulas in conjunctive normal form (CNF) is proposed. It extends the well-known graph representation of binary CNF formulas (2-SAT) to the general case. Every clause is represented as a set of (conditional) implications and encoded with different edges labeled with a set of literals, called context. This representation admits many interesting features. For example, a path from the node labeled with a literal ¬x to the node labeled with a literal x gives us an original way to compute the condition under which the literal x is implied. Using this representation, we show that classical resolution can be reformulated as a transitive closure on the generated graph. Interestingly enough, using the SAT graph-based representation three original applications are then derived. The first one deals with the 2-SAT strong backdoor set computation problem, whereas in the second one the underlying representation is used to derive hard SAT instances with respect to the state-of-the-art satisfiability solvers. Finally, a new preprocessing technique of CNF formulas which extends the well-known hyper-resolution rule is proposed. Experimental results show interesting improvements on many classes of SAT instances taken from the last SAT competitions.