On Adjacency Metric Dimension of Some Families of Graph

Abstract

Metric dimension of a graph is a well-studied concept. Recently, adjacency metric dimension of graph has been introduced. A set $Q_a\subset V\left(G\right)$ is considered to be an adjacency metric generator for $G$ if $u_1,u_2\in V\backslash Q_a$ (supposing each pair); there must exist a vertex $q\in Q_a$ along with the condition that $q$ is indeed adjacent to one of $u_1,u_2$ . The minimum number of elements in adjacency metric generator is the adjacency metric dimension of $G$ , denoted by $\text{di}{\text{m}}_a\left(G\right).$ In this work, we compute exact values of the adjacency metric dimension of circulant graph $C_n\left(1,2\right)$ , Möbius ladder, hexagonal Möbius ladder, and the ladder graph.