Graph Models of Cultural M-Grams: Polygonal Number Networks, Topological Index Partitions, and Geometric Transformations
Ümit Sarp
Source abstract
Star polygon motifs (m-grams), widely observed across historical architecture and combinatorial game theory, lie at the intersection of geometry and algebraic graph theory. In this study, we construct a formal graph family, denoted by Starm(n), by analyzing the topological structure of these motifs via the conceptual union of centered polygonal numbers (CPm) and triangular numbers (Tn). Its interior faces form an internally triangulated planar grid topology. After establishing the vertex and edge capacities via Euler’s planar formula, we obtain the exact edge partition space and the closed-form M-polynomial of the network. Applying the Deutsch–Klavžar differential operators to this polynomial, we derive closed and asymptotic expressions for key degree-based invariants: the First and Second Zagreb, Randić, Harmonic, Forgotten, Inverse Degree, and Sombor indices. All analytical derivations are validated computationally using breadth-first search (BFS) algorithms in linear time O(|V|+|E|). Finally, we introduce four geometric transformations—Annular Projection, Logarithmic Spiral Twisting, Girih Rosette Folding, and Kaleidoscopic Quotient Graphs—to classify the underlying generative mechanics of these cultural patterns.
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