On the Edge Metric Dimension of Certain Polyphenyl Chains

Abstract

The most productive application of graph theory in chemistry is the representation of molecules by the graphs, where vertices and edges of graphs are the atoms and valence bonds between a pair of atoms, respectively. For a vertex $w$ and an edge $f=c_1{c}_2$ of a connected graph $G$ , the minimum number from distances of $w$ with $c_1$ and $c_2$ is called the distance between $w$ and $f$ . If for every two distinct edges $f_1,f_2\in E\left(G\right)$ , there always exists $w_1\in W_E\subseteq V\left(G\right)$ such that $d\left(f_1,w_1\right)\ne d\left(f_2,w_1\right)$ , then $W_E$ is named as an edge metric generator. The minimum number of vertices in $W_E$ is known as the edge metric dimension of $G$ . In this paper, we calculate the edge metric dimension of ortho-polyphenyl chain graph $O_n$ , meta-polyphenyl chain graph $M_n$ , and the linear [n]-tetracene graph $T\left[n\right]$ and also find the edge metric dimension of para-polyphenyl chain graph $L_n$ . It has been proved that the edge metric dimension of $O_n$ , $M_n$ , and $T\left[n\right]$ is bounded, while $L_n$ is unbounded.