^{1, 2, 3}

^{2}

^{3, 4}

^{2}

^{1}

^{2}

^{3}

^{4}

The Kirchhoff index Kf(

It is well known that interconnection networks play an important role in parallel communication systems. An interconnection network is usually modelled by a connected graph

The adjacency matrix

Let

The Kirchhoff index has been attracting extensive attention due to its wide applications in physics, chemistry, graph theory, and so forth [

The main purpose of this paper is to investigate the Kirchhoff index of some combinatorial networks. The graph

The remainder of the paper is organized as follows. Section

In this section, we recall some underlying definitions and properties which we need to use in the proofs of our main results as follows.

The hypercube

The folded hypercube

Define the following operation of

Replace each vertex

There is an edge joining a vertex of

For each vertex

Recall the following two underlying conceptions that related to the above construction of

It is amazing and interesting that

Note that there is an elementary and important property: if

Yin and Wang [

For

M. Chen and B. X. Chen have studied the Laplacian spectra of

For

If

If

Let

Let

Let

Let

Let

For

Notice that

From Lemma

By virtue of Lemma

Replacing

Substituting (

From the definition graph

Combining (

It follows from (

This completes the proof.

The following theorem [

Let

Let

From Theorems

Theorem

In the following, we will further address the Kirchhoff index of

If

If

In an almost identical way as Theorem

For

In [

Let

Let

From Theorems

Theorems

To demonstrate the theoretical analysis, we provide some examples in this subsection, which are an application of our results. Without loss of generality, we suppose that the case is

By virtue of the definition of

Consequently, the same Kirchhooff index can be drawn as follows:

As the application of Theorem

Note that the eigenvalues of the Laplacian matrix of

Summing up the examples, the results above coincide the fact, which show our theorems are correct and effective.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The work of J. B. Liu was supported by Anhui Provincial Natural Science Foundation under Grant no. KJ2013B105 and the National Science Foundation of China under Grant nos. 11471016 and 11401004. The work of F. T. Hu was supported by Anhui Provincial Natural Science Foundation (1408085QA03) and the National Science Foundation of China under Grant no. 11401004. The authors would like to express their sincere gratitude to the anonymous referees for their valuable suggestions, which led to a significant improvement of the original paper.

^{m}-ary trees into hypercubes