Metric Dimension of Corona Product and Join Graph of Zero Divisor Graphs of Direct Product of Finite Fields

Abstract

The metric dimension of a graph is the smallest number of vertices such that all vertices are uniquely determined by their distances to the chosen vertices. The corona product of graphs G and H is the graph G ⊙ H obtained by taking one copy of G , called the center graph, | V ( G ) | copies of H , called the outer graph, and making the j t h vertex of G adjacent to every vertex of the j t h copy of H , where 1 ⩽ j ⩽ | V ( G ) | . The Join graph G + H of two graphs G and H is the graph with vertex set V ( G + H ) = V ( G ) ∪ V ( H ) and edge set E ( G + H ) = E ( G ) ∪ E ( H ) ∪ { u v : u ∈ V ( G ) , v ∈ V ( H ) } . In this paper, we determine the Metric dimension of Corona product and Join graph of zero divisor graphs of direct product of finite fields.