We introduce and study a class of simple graphs, the upper-maximal graphs (UM-graphs), associated to finite posets. The vertices of the UM-graph of a given poset P are the elements of P, and edges are formed by those vertices x and y whenever any maximal element of P that is greater than x is also greater than y or vise versa. We show that the class of UM-graphs constitutes a subclass of comparability graphs. We further provide a characterization of chordal UM-graphs, and compare UM-graphs with known bound graphs of posets.