On the edge metric dimension and Wiener index of the blow up of graphs

Abstract

Let $G=(V,E)$ be a connected graph. The distance between an edge $e=xy$ and a vertex $v$ is defined as $\T{d}(e,v)=\T{min}\{\T{d}(x,v),\T{d}(y,v)\}.$ A nonempty set $S \subseteq V(G)$ is an edge metric generator for $G$ if for any two distinct edges $e_1,e_2 \in E(G)$, there exists a vertex $s \in S$ such that $\T{d}(e_1,s) \neq \T{d}(e_2,s)$. An edge metric generating set with the smallest number of elements is called an edge metric basis of $G$, and the number of elements in an edge metric basis is called the edge metric dimension of $G$ and it is denoted by $\T{edim}(G)$. In this paper, we study the edge metric dimension of a blow up of a graph $G$, and also we study the edge metric dimension of the zero divisor graph of the ring of integers modulo $n$. Moreover, the Wiener index and the hyper-Wiener index of the blow up of certain graphs are computed.