On metric dimension of edge comb product of vertex-transitive graphs

Suppose finite graph G is simple, undirected and connected. If W is an ordered set of the vertices such that |W| = k, the representation of a vertex v is an ordered k-tuple consisting distances of vertex v with every vertices in W. The set W is defined as resolving vertex of G if the k-tuples of every two vertices are distinct. Metric dimension of G, which is denoted by dim(G), is the lowest size of W. In this paper, we provide a sharp lower bound of metric dimension for edge comb product graphs G∼ = T ▷e H where T is a tree graph and H is a vertex-transitive graph. Moreover, we determine the exact value of metric dimension for edge comb product graphs G ∼ =T ▷e Cin(1,2) where Cin(1,2) is a circulant graph.