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After a short introduction to the global supersymmetry in one dimension, presenting the basic definitions and tools, we construct the non-minimal linear representations of the N = 4 Extended Supersymmetry in one-dimension. They act on 8 bosonic and 8 fermionic fields. Inequivalent representations are specified by the mass-dimension of the fields, the connectivity of the associated graphs, commuting group and node choice group. Due to the important role that graphical representation plays for this classification, we investigate the correspondence between graphical representation, and algebraic representations. The distinction between the notion of equivalence for pure supermultiplets and the notion of equivalence for their associated graphs (Adinkras) is discussed. “Pure ” homogeneous linear supermultiplets (minimal and non-minimal) of the N = 4-Extended one-dimensional Supersymmetry Algebra are classified. “Pure ” means that they admit at least one graphical presentation (the corresponding graph/graphs are known as “Adinkras”). Discrete properties such as “chirality ” and “coloring ” can discriminate different su