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The resistance distance is a novel distance function on electrical network theory proposed by Klein and Randić. The Kirchhoff index Kf(

Throughout this paper we are concerned with finite undirected connected simple graphs (networks). Let

Given graphs

The resistance distances between vertices

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

Motivated by the above results, we present the corresponding calculating formulae for the Kirchhoff index of

In this section, we introduced some basic properties which we need to use in the proofs of our main results. Suppose that

Zhu et al. [

Let

The line graph of a graph

Let

A bipartite graph

Let

Let

Let

The following lemma gives an expression on

Let

Lemma

For the toroidal networks

Suppose the Laplacian eigenvalues of

According to Lemma

Since

The following consequence was presented in [

For the toroidal networks

By virtue of (

In the following theorem, we proposed the formula for calculating the Kirchhoff index of the line graph of

Let

Apparently the toroidal networks

We clearly obtained the following relationship

Substituting the results of Lemma

In an almost identical way as Theorem

Let

Noting that

Together with the results of Lemma

Now we proved the formula for estimating the Kirchhoff index in the total graph of

Let

Supposing that the Laplacian eigenvalues of

Applying Lemma

Notice that

Consequently, the relationships between

According to the results of Lemma

We will explore the formula for estimating the Kirchhoff index in the clique-inserted graph of

Let

Noting that

By virtue of Lemma

Obviously, it follows from Lemma

Based on (

It follows from (

Employing Lemma

The consequences of Lemma

We explore the asymptotic behavior of Kirchhoff index for the investigated networks above as

Let

According to (

The result is equivalent to

Let

Similarly, according to (

Let

Consider the summation term

Since

Combining with (

Let

From the proof of Theorem

As

Resistance distance was introduced by Klein and Randi

The asymptotic behavior of Kirchhoff indexes has been investigated with the applications of analysis approach, and the explicit approximate values are obtained by calculations for the related networks. The values of Kirchhoff indexes with respect to various networks can be immediately obtained via this approach; however, the quantities are rather difficult to calculate directly.

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

The work of Jia-Bao Liu is partly supported by the Natural Science Foundation of Anhui Province of China under Grant no. KJ2013B105. The work of Xiang-Feng Pan is partly supported by the National Science Foundation of China under Grant nos. 10901001, 11171097, and 11371028.