On the Anderson-Livingston graphs of 4-radical index of nilpotence finite completely primary rings

Edwin Nalianya Walubengo; Michael Onyango Ojiema; Fanuel Olege

doi:10.51867/ajernet.maths.7.3.1

Authors

Edwin Nalianya Walubengo Department of Mathematics, Masinde Muliro University of Science and Technology, Kenya https://orcid.org/0000-0002-6163-7246
Michael Onyango Ojiema Department of Mathematics, Masinde Muliro University of Science and Technology, Kenya https://orcid.org/0000-0001-9635-7597
Fanuel Olege Department of Mathematics, Masinde Muliro University of Science and Technology, Kenya

DOI:

https://doi.org/10.51867/ajernet.maths.7.3.1

Keywords:

Automorphism Group, Completely Primary Ring, Finite Ring, Galois Ring, Idealization, Zero Divisor Graph

Abstract

The classification of zero-divisor graphs of finite commutative rings is an active research area with connections to algebraic combinatorics. In this paper, we investigate the Anderson–Livingston graphs Γ(R) associated with completely primary finite rings R whose maximal ideal Z(R) satisfies (Z(R))⁴ = (0) but (Z(R))³ ≠ (0). Using the Raghavendran idealization procedure, we construct and characterize four classes of such rings with characteristics p, p², p³, and p⁴, explicitly describing the zero-divisor structure and radical filtration. For each class, we determine the basic graph invariants—diameter, girth, clique number, chromatic number, and degree sequences—showing that characteristics p, p², and p³ yield incomplete graphs of diameter 2, whereas characteristic p⁴ gives a complete graph. We further compute several advanced invariants, including the binding number, Zagreb indices, Wiener index, Mostar index, metric dimension, fractional metric dimension, edge differential, and domination number, for representative examples with p = 2 and r = 1, illustrated using TikZ drawings. In addition, we provide a complete classification of the automorphism groups Aut(Γ(R)) as products of symmetric groups and derive explicit orbit decompositions. These results extend the classification of zero-divisor graphs for higher radical indices and highlight the deep interplay between algebraic structure and graph theory.

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