Mostar index and edge Mostar index of polymers

Abstract

Let G=(V,E) be a graph and e=uv∈E. Define nu(e,G) be the number of vertices of G closer to u than to v. The number nv(e,G) can be defined in an analogous way. The Mostar index of G is a new graph invariant defined as Mo(G)=∑uv∈E(G)|nu(uv,G)−nv(uv,G)|. The edge version of Mostar index is defined as Moe(G)=∑e=uv∈E(G)|mu(e|G)−mv(G|e)|, where mu(e|G) and mv(e|G) are the number of edges of G lying closer to vertex u than to vertex v and the number of edges of G lying closer to vertex v than to vertex u, respectively. Let G be a connected graph constructed from pairwise disjoint connected graphs G1,…,Gk by selecting a vertex of G1, a vertex of G2, and identifying these two vertices. Then continue in this manner inductively. We say that G is a polymer graph, obtained by point-attaching from monomer units G1,…,Gk. In this paper, we consider some particular cases of these graphs that are of importance in chemistry and study their Mostar and edge Mostar indices.