The Unit Graphs (UGs) framework is a graph-based knowledge representation (KR) formalism that is designed to allow for the representation, manipulation, query, and reasoning over linguistic knowledge of the Explanatory Combinatorial Dictionary of the Meaning-Text Theory. One crucial advantage of this new formalism over other KR formalisms is that it is designed to represent valency-based predicates, and lexicographic definitions of lexical units in the form of semantic graphs. The goal of this paper is twofold. It both introduces the core of the UGs framework, and illustrates how it may be used to represent lexicographic definitions in the RELIEF lexicographic edition project.