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In this paper we discuss off-shell representations of N-extendedsupersymmetry in one dimension, ie, N-extended supersymmetric quantummechanics, and following earlier work on the subject codify them in terms ofcertain graphs, called Adinkras. This framework provides a method of generatingall Adinkras with the same topology, and so also all the correspondingirreducible supersymmetric multiplets. We develop some graph theoretictechniques to understand these diagrams in terms of a relatively small amountof information, namely, at what heights various vertices of the graph should be"hung".  We then show how Adinkras that are the graphs of N-dimensional cubes can beobtained as the Adinkra for superfields satisfying constraints that involvesuperderivatives. This dramatically widens the range of supermultiplets thatcan be described using the superspace formalism and organizes them. Othertopologies for Adinkras are possible, and we show that it is reasonable thatthese are also the result of constraining superfields using superderivatives.  The family of Adinkras with an N-cubical topology, and so also the sequenceof corresponding irreducible supersymmetric multiplets, are arranged in acyclical sequence called the main sequence. We produce the N=1 and N=2 mainsequences in detail, and indicate some aspects of the situation for higher N.Comment: LaTeX, 58 pages, 52 illustrations in color; minor typos correcte

ON GRAPH-THEORETIC IDENTIFICATIONS OF ADINKRAS, SUPERSYMMETRY REPRESENTATIONS AND SUPERFIELDS