This paper introduces implicational concept graphs as special existential concept graphs of power context families and shows how such implicational concept graphs give rise to a mathematical semantics of implications. The advantage of the offered mathematical semantics is that it opens the door to mathematical structure theory with its rich structural insights and patterns of argumentations. As a consequence, it could be proved that the implicational theory of implicational concept graphs is equivalent (in the main) to the theory of attribute implications of formal contexts. This result could even be generalized to an analogue result for clausal concept graphs.