Abstract

The ordered partition $\Pi=\{S_1,S_2,...,S_k\}$ of the vertices of the connected graph $G$ is a resolving partition, if for any vertex $x\in V$ with respect to the partition $\Pi$ is the vector $\zeta(x|\Pi)=(d(x,S_1),d(x,S_2),...,d(x,S_k))$ where $d(x,S_j), 1\leq j \leq k$ represents the distance between the vertex $x$ and the set $S_j$, is different for every pair of vertices and is denoted by $pd(G)$. The partition dimension is the minimum of $k$ for which there is a resolving partition. In this paper, we investigate the partition dimension of the extended zero divisor graphs of certain finite commutative rings.

Keywords and Phrases

Partition dimension, Extended zero divisor graph, Ring of integers, Commutative ring.

A.M.S. subject classification

13M05, 05C12, 13A70.

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