On the dominant local metric dimension of certain polyphenyl chain graphs

Abstract

The most practical application of graph theory in chemistry is to represent molecules as graphs, where the vertices represent atoms and the edges represent bonds between the atoms. Let G be a molecular graph with vertex set V(G) and edge set E(G). An atom (vertex) v resolves pair of adjacent vertices $v_1$ and $v_2$, if $d(v,v_1)\ne d(v,v_2)$, where $d(v,v_i)$ is the lenght of the shortest path joining v and $v_i$. An ordered set $W_l\subseteq V(G)$ is called a local resolving set, if $W_l$ resolves each pair of adjacent vertices of G. A subset $W\subseteq V(G)$ is said to be dominating set, if for each vertex $u\in V(G){\setminus } W$, there is a vertex $v\in W$, such that u is adjacent to v in G. An ordered set $W_l=\{w_1,w_2,w_3,...,w_k\}\subseteq V(G)$ is called dominant local resolving set, if $W_l$ is dominating set and also a local resolving set, and its minimum cardinality is known as dominant local metric dimension. In this article, the dominant local metric dimension of ortho-polyphenyl chain graph $O_n$, meta-polyphenyl chain graph $M_n$, para-polyphenyl chain graph $L_n$, and the linear [n]-tetracene chain graph T[n] are determined. Furthermore, the bond number, mobile bond order, free valence index, and total $\pi$-electron energy of these polyphenyl chain graphs are studied. These findings are helpful for the structure-activity relationship analysis of polyphenyl structures.