ON THE NILPOTENT DOT PRODUCT GRAPH OF A COMMUTATIVE RING

Document Type : Original Manuscript

Authors

Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India.

Abstract

Let B be a commutative ring with 1≠0, 1≤m<∞ be an integer and R=B×B×⋅⋅⋅×B (m times). In this paper, we introduce two types of (undirected) graphs, total nilpotent dot product graph denoted by TND(R) and nilpotent dot product graph denoted by ZND(R), in which vertices are from R∗=R∖{(0,0,...,0)} and ZN(R)∗ respectively, where ZN(R)∗={w∈R∗|wz∈N(R),for some z∈R∗}. Two distinct vertices w=(w1,w2,...,wm) and z=(z1,z2,...,zm) are said to be adjacent if and only if w⋅z∈N(B) (where w⋅z=w1z1+⋯+wmzm, denotes the normal dot product and N(B) is the set of nilpotent elements of B). We study about connectedness, diameter and girth of the graphs TND(R) and ZND(R). Finally, we establish the relationship between TND(R), ZND(R), TD(R) and ZD(R).

Keywords

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Journal of Algebraic Systems
Volume 13, Issue 2
July 2025
Pages 169-177

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