Metric and Upper Dimension of Extended Annihilating-Ideal Graphs

Abstract

The metric dimension problem is called navigation problem due to its application to robot navigation in space. Further this concept has wide applications in motion planning, sonar and loran station, and so on. In this paper, we study certain results on the metric dimension, upper dimension and resolving number of extended annihilating-ideal graph [Formula: see text] associated to a commutative ring [Formula: see text], denoted by [Formula: see text], [Formula: see text] and [Formula: see text], respectively. Here we prove the finiteness conditions of [Formula: see text] and [Formula: see text]. In addition, we characterize [Formula: see text], [Formula: see text] and [Formula: see text] for artinian rings and the direct product of rings.