This work is on the nature and properties of graphs which arise in the study of centered polygonal lacunary functions. Such graphs carry both graph-theoretic properties and properties related to the so-called p-sequences found in the study of centered polygonal lacunary functions. p-sequences are special bounded, cyclic sequences that occur at the natural boundary of centered polygonal lacunary functions at integer fractions of the primary symmetry angle. Here, these graphs are studied for their inherent properties. A ground-up set of planar graph construction schemes can be used to build the numerical values in p-sequences. Further, an associated three-dimensional graph is developed to provide a complementary viewpoint of the p-sequences. Polynomials can be assigned to these graphs, which characterize several important features. A natural reduction of the graphs original to the study of centered polygonal lacunary functions are called antipodal condensed graphs. This type of graph provides much additional insight into p-sequences, especially in regard to the special role of primes. The new concept of sprays is introduced, which enables a clear view of the scaling properties of the underling centered polygonal lacunary systems that the graphs represent. Two complementary scaling schemes are discussed.