^{1,2,3}

^{1,4}

^{2}

^{4}

^{1}

^{2}

^{3}

^{4}

The resistance distance between any two vertices of

In this work we are concerned with finite undirected connected simple graphs. For the graph theoretical definitions and notations we follow [

As an analogue to the Wiener index

Klein and Randić [

The Kirchhoff index has wide applications in physical interpretations, electric circuit, graph theory, and chemistry [

The hypercubes network

The hypercubes networks

The hypercubes networks

The hypercubes network

At the end of [

The remainder of this paper is organized as follows. Section

In this section, we recall some basic notations and results in graphs theory. The adjacency matrix

Yin and Wang [

For the hypercubes networks

Gutman and Mohar [

Let

Let

Let

It is worthwhile to note that the conclusion of Lemma 5 is not completely correct, the authors [

Let

The following Lemma give an expression on

Let

For proving the formula of the Kirchhoff index on the subdivision graph of hypercubes, we prove the following Lemma by utilizing Vieta’s Theorem; in our proof, some techniques in [

Let

Let Spec

Note that

In this section, we firstly give formula for the Kirchhoff index in the hypercubes

For the hypercubes networks

Since the hypercubes networks

Palacios and Renom studied the Kirchhoff index of the d-dimensional hypercube in [

The line graph of a graph

Let

Now, for convenience, we denoted the numbers of vertices and edges in the hypercubes networks

By Lemma

Comparing the spectrum of

We can easily obtain the spectrum of

Notice that the line graphs of hypercubes

Substituting the results of Theorem

The subdivision graph of a graph

Let

Now supposing that

Consequently, the coefficient of

Simplifying (

The total graph of a graph

Let

Let

In this paper, we focused on the Kirchhoff index of the hypercubes networks and related networks, which are important networks topology indexes for parallel processing computer systems. We obtained some exact formulae for the Kirchhoff index of the hypercubes networks

We also obtained the relationship for Kirchhoff index between hypercubes networks

Finally, the special formulae for the Kirchhoff indexes of

The work of Jinde Cao and Ahmed Elaiw was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University (KAU), under Grant 3–130/1434/HiCi. The authors, therefore, acknowledge technical and financial support of KAU; the work of Jiabao Liu was supported by the Natural Science Foundation of Anhui Province of China under Grant no. KJ2013B015; the work of Xiangfeng Pan was supported by the National Science Foundation of China under Grant nos. 10901001, 11171097, and 11371028.