On the Graph-Theoretic Properties of Compressed Zero Divisor Graphs of Product Rings over \(\Pi^n_{k=1}\) \(\mathbb{Z}_{pk}\)
S. G. Jakkewad *
Department of Mathematics, K. B. P. College Vashi, Navi Mumbai, India.
R. G. Metkar
Department of Mathematics, Indira Gandhi (Sr) College Cidco, Nanded, India.
*Author to whom correspondence should be addressed.
Abstract
This article explores the structural and graph-theoretic properties of the compressed zero-divisor graph of product rings R = \(\Pi^n_{k=1}\) \(\mathbb{Z}_{pk}\) where each pk is a prime and n ≥ 2. The study examines several key invariants of ΓC(R), including eccentricity, radius, diameter, girth, chromatic number, chromatic index, and clique number. It is shown that for n ≥ 3, ΓC(R) has radius 2, diameter 3, and contains cycles c\(l\) of every possible length \(l\) satisfying 3 ≤ \(l\) ≤ n. Moreover, it is proved that ω(ΓC(R)) = χ(ΓC(R)) = n, and the chromatic index follows χ(ΓC(R)) = 2n−1 − 1. These results provide deeper insights into the structural behaviour of compressed zero-divisor graphs associated with product rings over finite fields.
Keywords: Product ring, annihilator, compressed zero-divisor graph, girth, Clique n umber, Chromatic number