Maximum Reciprocal Degree Resistance Distance Index of Bicyclic Graphs

Gaixiang Cai,Xing-Xing Li,Guidong Yu,Junwei Wang

doi:10.1155/2021/9998763

Abstract

The reciprocal degree resistance distance index of a connected graph G is defined as RDR G = ∑ u , v ⊆ V G d G u + d G v / r G u , v , where r G u , v is the resistance distance between vertices u and v in G . Let ℬ n denote the set of bicyclic graphs without common edges and with n vertices. We study the graph with the maximum reciprocal degree resistance distance index among all graphs in ℬ n and characterize the corresponding extremal graph.

Highlights

Topological indices are numbers related to molecular structures, used as quantitative relationships between chemical structures and properties
Some of them are based on the distance of graph [1], the vertex degree [2], or the resistance distance. e first topological index was published by Wiener [3]
Alizadeh and Iranmanesh [11] proposed a new topological index, the reciprocal degree distance, which is defined as RDD(G) 1

Summary

Introduction

Topological indices are numbers related to molecular structures, used as quantitative relationships between chemical structures and properties. Alizadeh and Iranmanesh [11] proposed a new topological index, the reciprocal degree distance, which is defined as RDD(G) 1. Analogous to the reciprocal degree distance index RDD(G), Cai et al [14] introduced a new graph invariant based on both the vertex degree and the resistance, named the reciprocal degree resistance distance index, shown as follows: RDR(G) 􏽘 dG(u) + dG(v). We determine the graph with the maximum reciprocal degree resistance distance index among all graphs in Bn and characterize the corresponding extremal graph. Is paper is organized as follows: in the second part, we give three types of transformation, edge-lifting transformation, cycle-lifting transformation, and cycle-shrinking transformation, to keep the reciprocal degree resistance distance index increasing. We give the maximum reciprocal degree resistance distance index among all graphs in Bn

Preliminaries

Maximum Reciprocal Degree Resistance Distance Index of Bicyclic Graphs

Conclusions