A Unified Cyclotomic–Laplacian Spectrum of Circular Graphs: Resultant Obstructions, Intrinsic Divisors, and Primitive Prime Divisors

The provided text is a mathematical research draft that rigorously investigates the cyclotomic-Laplacian spectra of odd cycle graphs by connecting their structural eigenvalues to classical cyclotomic polynomials and Lucas-sequence evaluations. The author systematically reconstructs and corrects previous technical claims of their earlier work, establishing vital algebraic tools such as spectral factorization, resultant identities, and exact prime-rank congruences over finite fields. A central achievement of the paper is proving an elementary growth bound that guarantees the existence of primitive prime divisors outside the standard spectral band, while carefully accounting for necessary exceptions. Ultimately, the work serves as a transparent and corrected mathematical framework that bridges graph theory, number theory, and classical polynomial sequences. A Unified Cyclotomic–Laplacian Spectrum of Circular Graphs Short Overview This paper investigates the spectral properties of odd cycle graph Laplacians and bridges graph theory with classical number theory. By factoring the reduced Laplacian polynomial into irreducible components indexed by odd divisors, the work establishes how graph eigenvalues map to cyclotomic polynomials. It provides a corrected law of apparition for prime factors, bounds non-primitive (intrinsic) divisors, and proves the existence of primitive prime divisors outside the main spectral band . Core Precepts and Connected Topics Spectral Factorization & Cyclotomic Transfer: Shows how nonzero eigenvalues of odd cycle graphs factor into irreducible polynomials, translating directly to standard cyclotomic polynomials via a simple variable shift. Resultant Identities & Value-Wise Coprimality: Applies Apostol–Lehmer resultant formulas to spectral coordinates, proving that distinct spectral components share no common integer factors unless their indices have a prime-power relationship. Law of Apparition & Finite Fields: Identifies exact multiplicative order conditions in finite field quadratic extensions that dictate when and how prime factors divide spectral polynomial values. Intrinsic Divisors & Primitive Prime Bounds: Sets strict bounds on non-primitive prime factors and proves that primitive prime divisors of exact rank exist outside the spectral band (except for index 5 at evaluation 5) . Fibonacci, Lucas, and Jacobi Evaluations: Connects spectral evaluations at boundary points directly to Fibonacci numbers, Lucas numbers, and the Jacobi quadratic reciprocity symbol [5, 13]. Keywords Graph Spectra, Cycle Graphs, Graph Laplacian, Cyclotomic Polynomials, Lucas Sequences, Fibonacci Numbers, Primitive Prime Divisors, Resultants, Law of Apparition, Finite Fields, Quadratic Reciprocity Classifications (MSC 2020) * 05C50 (Graph Spectra & Linear Algebra) * 11B39 (Fibonacci and Lucas Sequences) * 11R18 (Cyclotomic Fields) * 11A41 (Primes)