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=c1c2 of a connected graph G, the minimum number from distances of w with c1 and c2 is called the distance between w and f. If for every two distinct edges f1,f2∈EG, there always exists w1∈WE⊆VG such that df1,w1≠df2,w1, then WE is named as an edge metric generator. The minimum number of vertices in WE is known as the edge metric dimension of G. In this paper, we calculate the edge metric dimension of ortho-polyphenyl chain graph On, meta-polyphenyl chain graph Mn, and the linear [n]-tetracene graph Tn and also find the edge metric dimension of para-polyphenyl chain graph Ln. It has been proved that the edge metric dimension of On, Mn, and Tn is bounded, while Ln is unbounded.