PARTITION DIMENSION OF EXTENDED ZERO DIVISOR GRAPHS

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.