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 $G \cong T$ ▷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 $G \cong T$ ▷e $Ci_n(1,2)$ where $Ci_n(1,2)$ is a circulant graph.