Resistance distance of generalized wheel and dumbbell graph using symmetric {1}-inverse of Laplacian matrix

Abstract

A new class of graphs called dumbbell graphs, denoted by DB(Wm,n) is the graph obtained from two copies of generalized wheel graph Wm,n, m ≥ 2, n ≥ 3. It is a graph on 2 (m + n) vertices obtained by connecting m-vertices in one copy with the corresponding vertices in the other copy. The resistance distance between two vertices vi and vj, denoted by rij , is defined as the effective electrical resistance between them if each edge of G is replaced by 1 ohm resistor. The Kirchhoff index is the sum of the resistance distances between all pairs of vertices in the graph. In this paper, we formulate the resistance distance of Wm,n and DB(Wm,n) using Symmetric {1}-inverse of Laplacian matrices. We provide examples to illustrate the proposed method and also obtain the Kirchhoff indices for these examples.