On the resistance diameters of graphs and their line graphs

Autor: Xiang-Feng Pan, Yun-Xiang Li, Si-Ao Xu, Hongbo Hua
Rok vydání: 2022
Předmět:
Zdroj: Discrete Applied Mathematics. 306:174-185
ISSN: 0166-218X
DOI: 10.1016/j.dam.2021.09.033
Popis: Let G = ( V ( G ) , E ( G ) ) be a graph with vertex set V ( G ) and edge set E ( G ) . The line graph L G of G is the graph with E ( G ) as its vertex set and two vertices of L G are adjacent in L G if and only if they have a common end-vertex in G . The resistance distance R G ( x , y ) between two vertices x , y of G is equal to the effective resistance between the two vertices in the corresponding electrical network in which each edge of G is replaced by a unit resistor. The resistance diameter D r ( G ) of G is the maximum resistance distance among all pairs of vertices of G . In this paper, it was shown that the resistance diameter of the line graph of a tree or unicyclic graph is no more than that of its initial graph by utilizing series and parallel principles, the principle of elimination and star-mesh transformation in electrical network theory. And experiment also indicated that the inequality D r ( L G ) ≤ D r ( G ) is true for every simple nonempty connected graph G with less than 12 vertices. Thus it was conjectured that D r ( L G ) ≤ D r ( G ) for every simple nonempty connected graph G .
Databáze: OpenAIRE
Externí odkaz: