Triangular Root: A Solution to Three Conjectures and a Bridge Between c-Cyclic Graphs and Ramanujan τ Numbers

For every c≥3 and n≥2c−2, we prove that the unique minimum degree sequence for the inverse degree index I(G) and the symmetric division deg index SDD(G) is exactly [32c−2,2n−2c+2]. For the maximum, we give explicit counterexamples for small orders and prove that for sufficiently large n both indices are simultaneously maximized by the same optimal degree sequence, determined by the triangular root 8c+1−12. We resolve three known conjectures: Palacios’ conjecture is confirmed for the minimum and refuted for the maximum for small orders, while it is asymptotically confirmed for large orders; Bianchi et al.’s conjecture is fully proved; Ali et al.’s conjecture is confirmed for the maximum (asymptotically) and refuted in its strict form for the minimum, though the degree sequence is the same. Additionally, the same triangular root is connected to Ramanujan τ numbers, where canonical threshold graphs give a combinatorial representation of the τ-function. The Defect Lemma τ(p2)−τ(p)2=−p11 is a well-known consequence of Hecke multiplicativity, first proved by Mordell. In this paper, we provide new elementary proofs of the Defect Lemma for p=3 and p=5 that do not rely on the theory of modular forms or on Mordell’s analytic methods. These proofs are based on q-series identities and recurrence formulas, and are motivated by the combinatorial representation of τ(n) via canonical threshold graphs.