Many problems in economics can be formulated as early constrained mathematical optimization problems, where the feasible solution set X represents a convex polyhedral set. In practice, the set X frequently contains degenerate vertices, yielding diverse problems in the determination of an optimal solution as well as in postoptimal analysis. Degeneracy graphs represent a useful tool for describing and solving degeneracy problems. The study of degeneracy graphs opens a new field of research with many theoretical aspects and practical applications. The present monograph pursues two aims. On the one hand, the theory of degeneracy graphs is developed generally, to serve as a basis for further applications. On the other hand, degeneracy graphs are used to explain simplex cycling, and necessary and sufficient conditions for cycling are derived.