On the resistance diameters of graphs and their line graphs

Abstract

Let G=(V(G),E(G)) be a graph with vertex set V(G) and edge set E(G). The line graph LG of G is the graph with E(G) as its vertex set and two vertices of LG are adjacent in LG if and only if they have a common end-vertex in G. The resistance distance RG(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 Dr(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 Dr(LG)≤Dr(G) is true for every simple nonempty connected graph G with less than 12 vertices. Thus it was conjectured that Dr(LG)≤Dr(G) for every simple nonempty connected graph G.