Mostar index and edge Mostar index of polymers

Abstract

Let $G=(V,E)$ be a graph and $e=uv\in E$. Define $n_u(e,G)$ be the number of vertices of $G$ closer to $u$ than to $v$. The number $n_v(e,G)$ can be defined in an analogous way. The Mostar index of $G$ is a new graph invariant defined as $Mo(G)=\sum_{uv\in E(G)}|n_u(uv,G)-n_v(uv,G)|$. The edge version of Mostar index is defined as $Mo_e(G)=\sum_{e=uv\in E(G)} |m_u(e|G)-m_v(G|e)|$, where $m_u(e|G)$ and $m_v(e|G)$ 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 $G_1,\ldots ,G_k$ by selecting a vertex of $G_1$, a vertex of $G_2$, 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 $G_1,...,G_k$. 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.