Some Results Of Resistance Distance And The Kirchhoff Index In Double Graphs

The Kirchhoff index of a graph G is de?ned as Kf(G) =12∑n i=1∑n j=1r(vi, vj), where r(vi, vj) is the resistance distance between viand vj. Let G be a simple graph, DG represents the double graph of G. In this paper, ?rst, we get a relation between the Kirchhoff index of the double graph and the Kirchhoff index of the original graph by means of Laplacian spectrum: Kf(DG) = n∑n i=11d(vi)+ Kf(G). Then, by GTS(generalized tree shift), we determine the trees with the ?rst three maximum and minimum values of Kf(DT). Next,we derive a closed form for resistance distance of DG in terms of that of G by an algorithm of computing resistance distance, then we get the general expressions of resistance distance and Kirchhoff index of k-iterated double graph respectively, and we also consider the asymptotic behavior of Kirchhoff index of k-iterated double graphs. Finally, as applications, we give the speci?c expressions of resistance distances and Kirchhoff indices of several kinds of k-iterated double graphs(such as, the complete graph, the tree, the cycle, etc).