On the resistance diameters of graphs and their line graphs.

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. [ABSTRACT FROM AUTHOR]