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.