The Kirchhoff Index of Hypercubes and Related Complex Networks

Abstract

The resistance distance between any two vertices ofGis defined as the network effective resistance between them if each edge ofGis replaced by a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all the pairs of vertices inG. We firstly provided an exact formula for the Kirchhoff index of the hypercubes networksQnby utilizing spectral graph theory. Moreover, we obtained the relationship of Kirchhoff index between hypercubes networksQnand its three variant networksl(Qn),s(Qn),t(Qn)by deducing the characteristic polynomial of the Laplacian matrix related networks. Finally, the special formulae for the Kirchhoff indexes ofl(Qn),s(Qn), andt(Qn)were proposed, respectively.