A family G of connected graphs is said to be a family with constant metric dimension if its metric dimension is finite and does not depend upon the choice of G in G. In this paper, we study the metric dimension of the generalized Petersen graphs P(2m, m − 1) and give a partial answer to an open problem raised in [13]: Is the generalized Petersen graphs P(s, t), for s ≥ 7 and 3 ≤ t ≤ , a family of graphs with constant metric dimension? We prove that the generalized Petersen graphs P(2m, m − 1) have metric dimension equal to 3 for all odd m ≥ 3, and equal to 4 for all even m ≥ 4.

This paper is devoted to the study of the arithmetic graph of a composite number m, denoted by ${\mathcal{A}}_m$ . It has been observed that there exist different composite numbers for which the arithmetic graphs are isomorphic. It is proved that the maximum distance between any two vertices of ${\mathcal{A}}_m$ is two or three. Conditions under which the vertices have the same degrees and neighborhoods have also been identified. Symmetric behavior of the vertices lead to the study of the metric dimension of ${\mathcal{A}}_m$ which gives minimum cardinality of vertices to distinguish all vertices in the graph. We give exact formulae for the metric dimension of ${\mathcal{A}}_m$ , when m has exactly two distinct prime divisors. Moreover, we give bounds on the metric dimension of ${\mathcal{A}}_m$ , when m has at least three distinct prime divisors.