Since the 1990s, as we know, comparability and incomparability graphs find use in experimental sciences, particularly in chemistry and physical chemistry as well as in statistical physics, to characterize molecular properties, order isomers or specify the topology of certain structures such as chemical structures. In this mostly historical article, readable by a non-specialist, we recall first some basic notions of graph theory. Then, we define the notions of comparability and incomparability graphs of a poset, summarize the recent advances on the question and compute the number of edges of these graphs, using in particular Boolean representation of posets. We also give a special method in the case of Boolean lattices.