Annihilator Depth Partitions and Spectral Reduction of Zero–Divisor Graphs over Truncated Polynomial Rings

Author(s)	Dr. Hanumantha Reddy D T, Dr. Harish K C
Country	India
Abstract	Zero–divisor graphs encode annihilation relations in commutative rings through a combinatorial framework. A bivariate annihilator depth invariant is introduced for truncated polynomial rings over finite chain rings, with particular focus on S=Z_(p^k ) [x]/(x^n). This invariant simultaneously captures p–adic valuation and polynomial nilpotency, inducing a canonical depth partition of the nonzero zero–divisors. It is shown that this partition yields a complete multipartite structure of the zero–divisor graph Γ(S) and is equitable. As a consequence, the adjacency spectrum of Γ(S) reduces exactly to the spectrum of an associated quotient matrix. Applications include the equality χ(Γ(S)) = ω(Γ(S)), explicit spectral bounds, and a uniform bound on the diameter of Γ(S). Representative examples illustrate the framework, and extensions to truncated polynomial rings over arbitrary finite chain rings are indicated.
Keywords	Zero–divisor graph; annihilator depth; truncated polynomial ring; equitable partition; spectral graph theory.
Field	Mathematics
Published In	Volume 8, Issue 4, July-August 2026
Published On	2026-07-09
DOI	https://doi.org/10.36948/ijfmr.2026.v08i04.83283

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