Abstract

A vertex-degree-based (VDB, for short) topological index $\phi$ is defined for a graph G with n vertices as

$\phi (G)=\sum _{1\le i\le j\le n-1}{m}_{i,j}(G)\phi (i,j),$

where $m_{i,j}=m_{i,j}(G)$ is the number of edges of G joining a vertex of degree i with a vertex of degree j, and $\left\{\phi (i,j)\right\}$ is a set of real numbers. In this paper, we present a general formula to calculate VDB topological indices of graphene systems. As a byproduct, we solve the following extremal value problem: given a VDB topological index $\phi ,$ find the graphene with maximal value of $\phi$ and the graphene with minimal value of $\phi ,$ among all graphene with h hexagons.

中文翻译：

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基于顶点度数的石墨烯拓扑指数

摘要

基于顶点度数（VDB，简称 VDB）的拓扑索引$\phi$为具有n个顶点的图G定义为

$\phi (G)=\sum _{1\le \mathrm{一世}\le j\le n-1}{米}_{\mathrm{一世},j}(G)\phi (\mathrm{一世},j),$

在哪里${米}_{\mathrm{一世},j}={米}_{\mathrm{一世},j}(G)$是连接度为i的顶点与度为j的顶点的G的边数，并且$\left\{\phi (\mathrm{一世},j)\right\}$是一组实数。在本文中，我们提出了一个计算石墨烯系统 VDB 拓扑指数的通用公式。作为副产品，我们解决了以下极值问题：给定一个 VDB 拓扑索引$\phi ,$找到最大值为的石墨烯$\phi$和具有最小值的石墨烯$\phi ,$在所有具有h六边形的石墨烯中。