Further Results on Resistance Distance and Kirchhoff Index in Electric Networks

Abstract

In electric circuit theory, it is of great interest to compute the effective resistance between any pairs of vertices of a network, as well as the Kirchhoff index. Let Q(G) be the graph obtained from G by inserting a new vertex into every edge of G and by joining by edges those pairs of these new vertices which lie on adjacent edges of G. The set of such new vertices is denoted by I(G). The Q-vertex corona of G1 and G2, denoted by G1⊙QG2, is the graph obtained from vertex disjoint Q(G1) and VG1 copies of G2 by joining the ith vertex of V(G1) to every vertex in the ith copy of G2. The Q-edge corona of G1 and G2, denoted by G1⊖QG2, is the graph obtained from vertex disjoint Q(G1) and IG1 copies of G2 by joining the ith vertex of I(G1) to every vertex in the ith copy of G2. The objective of the present work is to obtain the resistance distance and Kirchhoff index for composite networks such as Q-vertex corona and Q-edge corona networks.