The Kirchhoff Index of Toroidal Meshes and Variant Networks

Abstract

The resistance distance is a novel distance function on electrical network theory proposed by Klein and Randić. The Kirchhoff index Kf(𝐺) is the sum of resistance distances between all pairs of vertices in 𝐺. In this paper, we established the relationships between the toroidal meshes network 𝑇_𝑚×𝑛 and its variant networks in terms of the Kirchhoff index via spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes of 𝐿(𝑇_𝑚×𝑛), 𝑆(𝑇_𝑚×𝑛), 𝑇(𝑇_𝑚×𝑛), and 𝐶(𝑇_𝑚×𝑛) were proposed, respectively. Finally, the asymptotic behavior of Kirchhoff indexes in those networks is obtained by utilizing the applications of analysis approach.

1. Introduction

Throughout this paper we are concerned with finite undirected connected simple graphs (networks). Let 𝐺=(𝑉,𝐸) be a graph with vertices labelled 1,2,…,𝑛. The adjacency matrix 𝐴(𝐺) of 𝐺 is an 𝑛×𝑛 matrix with the (𝑖,𝑗)-entry equal to 1 if vertices 𝑖 and 𝑗 are adjacent and 0 otherwise. Suppose 𝐷(𝐺)=diag(𝑑₁(𝐺),𝑑₂(𝐺),…,𝑑_𝑛(𝐺)) is the degree diagonal matrix of 𝐺, where 𝑑_𝑖(𝐺) is the degree of the vertex 𝑖, 𝑖=1,2,…,𝑛. Let 𝐿(𝐺)=𝐷(𝐺)−𝐴(𝐺) be called the Laplacian matrix of 𝐺. Then, the eigenvalues of 𝐴(𝐺) and 𝐿(𝐺) are called eigenvalues and Laplacian eigenvalues of 𝐺, respectively.

Given graphs 𝐺 and 𝐻 with vertex sets 𝑈 and 𝑉, the Cartesian product 𝐺□𝐻 of graphs 𝐺 and 𝐻 is a graph such that the vertex set of 𝐺□𝐻 is the Cartesian product 𝑈×𝑉; and any two vertices (𝑢,𝑢^′) and (𝑣,𝑣^′) are adjacent in 𝐺□𝐻 if and only if either 𝑢=𝑣 and 𝑢^′ is adjacent with 𝑣^′ in 𝐻 or 𝑢^′=𝑣^′ and 𝑢 is adjacent with 𝑣 in 𝐺 [1]. It is well known that many of the graphs (networks) operations can produce a great deal of novel types of graphs (networks), for example, Cartesian product of graphs, line graph, subdivision graph, and so on. The clique-inserted graph, denoted by 𝐶(𝐺), is defined as a line graph of the subdivision graph 𝑆(𝐺) [2, 3]. The subdivision graph of an 𝑟-regular graph is (𝑟,2)-semiregular graph. Consequently, the clique-inserted graph of an 𝑟-regular graph is the line graph of an (𝑟,2)-semiregular graph.

The resistance distances between vertices 𝑖 and 𝑗, denoted by 𝑟_𝑖𝑗, are defined as the effective electrical resistance between them if each edge of 𝐺 is replaced by a unit resistor [4]. A famous distance-based topological index, the Kirchhoff index Kf(𝐺), is defined as the sum of resistance distances between all pairs of vertices in 𝐺; that is, Kf(𝐺)=(1/2)∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝑟_𝑖𝑗(𝐺), known as the Kirchhoff index of 𝐺 [4]; recently, this classical index has also been interpreted as a measure of vulnerability of complex networks [5].

The Kirchhoff index attracted extensive attention due to its wide applications in physics, chemistry, graph theory, and so forth [6–13]. Besides, the Kirchhoff index also is a structure descriptor [14]. Unfortunately, it is rather hard to directly design some algorithms [15–17] to calculate resistance distances and the Kirchhoff indexes of graphs. So, many researchers investigated some special classes of graphs [18–21]. In addition, many efforts were also made to obtain the Kirchhoff index bounds for some graphs [17, 22]. Details on its theory can be found in recent papers [17, 22] and the references cited therein.

Motivated by the above results, we present the corresponding calculating formulae for the Kirchhoff index of 𝐿(𝑇_𝑚×𝑛), 𝑆(𝑇_𝑚×𝑛), 𝑇(𝑇_𝑚×𝑛), and 𝐶(𝑇_𝑚×𝑛) in this paper. The rest of this paper is organized as follows. Section 2 presents some underlying notations and preliminaries in our discussion. The proofs of our main results and some asymptotic behavior of Kirchhoff index are proposed in Sections 3 and 4, respectively.

2. Notations and Some Preliminaries

In this section, we introduced some basic properties which we need to use in the proofs of our main results. Suppose that 𝑇_𝑚×𝑛 stands for the graphs 𝐶_𝑚□𝐶_𝑛 for the convenience of description. It is trivial for 𝑚,𝑛 are 1, 2, without loss of generality, we discuss the situations for any positive integer 𝑚,𝑛≥3.

Zhu et al. [15] and Gutman and Mohar [8] proved the relations between Kirchhoff index of a graph and Laplacian eigenvalues of the graph as follows.

Lemma 1 (see [8, 15]). Let 𝐺 be a connected graph with 𝑛≥2 vertices and let 𝜇₁≥𝜇₂≥⋯≥𝜇_𝑛=0 be the Laplacian eigenvalues of graph 𝐺; then 𝐾𝑓(𝐺)=𝑛^𝑛−1_𝑖=11𝜇_𝑖. (1)

The line graph of a graph 𝐺, denoted by 𝐿(𝐺), is the graph whose vertices correspond to the edges of 𝐺 with two vertices of 𝐿(𝐺) being adjacent if and only if the corresponding edges in 𝐺 share a common vertex. The subdivision graph of a graph 𝐺, denoted by 𝑆(𝐺), is the graph obtained by replacing every edge in 𝐺 with a copy of 𝑃₂ (path of length two). The total graph of a graph 𝐺, denoted by 𝑇(𝐺), is the graph whose vertices correspond to the union of the set of vertices and edges of 𝐺, with two vertices of 𝑇(𝐺) being adjacent if and only if the corresponding elements are adjacent or incident in 𝐺. Let 𝑃_𝐺(𝑥) be the characteristic polynomial of the Laplacian matrix of a graph 𝐺; the following results were shown in [23].

Lemma 2 (see [23]). Let 𝐺 be an 𝑟-regular connected graph with 𝑛 vertices and 𝑚 edges; then𝑃_𝐿(𝐺)(𝑥)=(𝑥−2𝑟)^𝑚−𝑛𝑃_𝐺(𝑥),𝑃_𝑆(𝐺)(𝑥)=(−1)^𝑚(2−𝑥)^𝑚−𝑛𝑃_𝐺(𝑥(,𝑃𝑟+2−𝑥))_𝑇(𝐺)(𝑥)=(−1)^𝑚(𝑟+1−𝑥)^𝑛(2𝑟+2−𝑥)^𝑚−𝑛𝑃_𝐺𝑥(𝑟+2−𝑥),𝑟−𝑥+1 (2) where 𝑃_𝐿(𝐺)(𝑥), 𝑃_𝑆(𝐺)(𝑥), and 𝑃_𝑇(𝐺)(𝑥) are the characteristic polynomials for the Laplacian matrix of graphs 𝐿(𝐺), 𝑆(𝐺), and 𝑇(𝐺), respectively.

A bipartite graph 𝐺 with a bipartition 𝑉(𝐺)=(𝑈,𝑉) is called an (𝑟,𝑠)-semiregular graph if all vertices in 𝑈 have degree 𝑟 and all vertices in 𝑉 have degree 𝑠. Apparently, the subdivision graph of an 𝑟-regular-graph 𝐺 is (𝑟,2)-semiregular graph.

Lemma 3 (see [24]). Let 𝐺 be an (𝑟,𝑠)-semiregular connected graph with 𝑛 vertices. Then 𝑃_𝐿(𝐺)(𝑥)=(−1)^𝑛(𝑥−(𝑟+𝑠))^𝑚−𝑛𝑃_𝐺(𝑟+𝑠−𝑥), (3) where 𝑃_𝐿(𝐺)(𝑥) is the Laplacian characteristic polynomial of the line graph 𝐿(𝐺) and 𝑚 is the number of edges of 𝐺.

Lemma 4 (see [23]). Let 𝐺 be a connected simple r-regular graph with 𝑛 vertices and 𝑚 edges and let 𝐿(𝐺) be the line graph of 𝐺. Then (𝐿(𝐺=𝑟𝐾𝑓))2(𝐺)+1𝐾𝑓4𝑛(𝑚−𝑛). (4)

Lemma 5 (see [23]). Let 𝐺 be a connected simple 𝑟-regular graph with 𝑛≥2 vertices; then (𝑆(𝐺=()𝐾𝑓))𝑟+2²2(𝐺)+𝑟𝐾𝑓²−4𝑛²+4𝑛8. (5)

The following lemma gives an expression on 𝐾𝑓(𝑇(𝐺)) and 𝐾𝑓(𝐺) of a regular graph 𝐺.

Lemma 6 (see [25]). Let 𝐺 be a 𝑟-regular connected graph with 𝑛 vertices and 𝑚 edges, and 𝑟≥2; then (𝑇(𝐺=𝑛()𝐾𝑓))𝑟+2)(𝑟+42(𝑟+3)^𝑛−1_𝑖=11𝜇_𝑖+()+3+𝑟𝑟+2²2()(𝐺)+𝑛𝑟+3𝐾𝑓²𝑟²−48()+𝑛𝑟+12. (6)

Lemma 7 presents the formula for calculating Kirchhoff index of 𝑇_𝑚×𝑛; in the following proof, some techniques in [26] are referred to.

Lemma 7 (see [26]). For the toroidal networks 𝑇_𝑚×𝑛 with any positive integer 𝑚,𝑛≥3, 𝐾𝑓𝑇_𝑚×𝑛=𝑚𝑛^𝑚−1_𝑖=1^𝑛−1_𝑗=114sin²(𝑖𝜋/𝑚)+4sin²𝑚𝑗𝜋/𝑛+𝑛³−𝑚𝑛12+𝑚³−𝑛.12 (7)

Proof. Suppose the Laplacian eigenvalues of 𝐶_𝑚 and 𝐶_𝑛 are 4sin²(𝑖𝜋/𝑚) and 4sin²(𝑗𝜋/𝑛), 𝑖=0,1,…,𝑚−1; 𝑗=0,1,…,𝑛−1; then the Cartesian product 𝐺□𝐻 and the Laplacian eigenvalues of 𝐿_{(𝑐𝑚□𝐶𝑛)} are 4sin²𝑖𝜋𝑚+4sin²𝑗𝜋𝑛,𝑖=0,1,…,𝑚−1;𝑗=0,1,…,𝑛−1. (8)
According to Lemma 1, the Kirchhoff index of the toroidal networks 𝐾𝑓(𝑇_𝑚×𝑛) is 𝐾𝑓𝑇_𝑚×𝑛=𝑚𝑛_{(𝑖,𝑗)∈𝐴×𝐵⧵{(0,0)}}14sin²(𝑖𝜋/𝑚)+4sin²,{}{}𝑗𝜋/𝑛𝐴=0,1,…,𝑚−1,𝐵=0,1,…,𝑛−1=𝑚𝑛^𝑚−1_𝑖=0^𝑛−1_𝑗=014sin²(𝑖𝜋/𝑚)+4sin²,≠()𝑗𝜋/𝑛𝑖,𝑗0,0=𝑚𝑛^𝑚−1_𝑖=114sin²()+𝑖𝜋/𝑚^𝑛−1_𝑗=114sin²+𝑗𝜋/𝑛_{𝑚−1𝑖=1𝑛−1𝑗=1}14sin²(𝑖𝜋/𝑚)+4sin²𝑚𝑗𝜋/𝑛=𝑛^𝑚−1_𝑖=114sin²()𝑛𝑖𝜋/𝑚+𝑚^𝑛−1_𝑗=114sin²𝑗𝜋/𝑛+𝑚𝑛^𝑚−1_𝑖=1^𝑛−1_𝑗=114sin²(𝑖𝜋/𝑚)+4sin²𝐶𝑗𝜋/𝑛=𝑛𝐾𝑓_𝑚+𝑚𝐾𝑓𝐶_𝑛+𝑚𝑛^𝑚−1_𝑖=1^𝑛−1_𝑗=114sin²(𝑖𝜋/𝑚)+4sin²𝑗𝜋/𝑛=𝑚𝑛^𝑚−1_𝑖=1^𝑛−1_𝑗=114sin²(𝑖𝜋/𝑚)+4sin²𝑚𝑗𝜋/𝑛+𝑛³−𝑚𝑛12+𝑚³−𝑛.12 (9)(10)
Since 𝐾𝑓(𝐶_𝑛)=(𝑛³−𝑛)/12, (10) in the last line holds.
The following consequence was presented in [26]. Here we give a short proof.

Lemma 8 (see [26]). For the toroidal networks 𝑇_𝑚×𝑛 with any positive integer 𝑚,𝑛≥3, lim_𝑚→∞lim_𝑛→∞𝐾𝑓𝑇_𝑚×𝑛𝑚²𝑛²≈1.905. (11)

Proof. By virtue of (9), one can derive that 𝐾𝑓𝑇_𝑚×𝑛=𝑚𝑛^𝑚−1_𝑖=0^𝑛−1_𝑗=014sin²(𝑖𝜋/𝑚)+4sin²,≠().𝑗𝜋/𝑛𝑖,𝑗0,0 (12) Hence, lim_𝑚→∞lim_𝑛→∞𝐾𝑓𝑇_𝑚×𝑛𝑚²𝑛²=lim_𝑚→∞lim_𝑛→∞1𝑚𝑛^𝑚−1_𝑖=0^𝑛−1_𝑗=01(),≠()=14−2cos2𝜋𝑖/𝑚−2cos2𝜋𝑗/𝑛𝑖,𝑗0,012𝜋2𝜋₀^2𝜋₀^2𝜋𝑑_𝑥𝑑_𝑦4−2cos𝑥−2cos𝑦≈1.905. (13)

3. Main Results

3.1. The Kirchhoff Index of 𝐿(𝑇_𝑚×𝑛)

In the following theorem, we proposed the formula for calculating the Kirchhoff index of the line graph of 𝑇_𝑚×𝑛, denoted by Kf(𝐿(𝑇_𝑚×𝑛)).

Theorem 9. Let 𝐿(𝑇_𝑚×𝑛) be line graphs of 𝑇_𝑚×𝑛 with any positive integer 𝑚,𝑛≥3; then 𝐾𝑓𝐿𝑇_𝑚×𝑛=2𝑚𝑛^𝑚−1_𝑖=1^𝑛−1_𝑗=114sin²(𝑖𝜋/𝑚)+4sin²𝑚𝑗𝜋/𝑛+𝑛³−𝑚6𝑛+𝑚³−𝑛6+𝑚²𝑛²4. (14)

Proof. Apparently the toroidal networks 𝑇_𝑚×𝑛 are 4-regular graphs which have 𝑚𝑛 vertices and 2𝑚𝑛 edges, respectively.
We clearly obtained the following relationship 𝐾𝑓(𝐿(𝑇_𝑚×𝑛)) and 𝐾𝑓(𝑇_𝑚×𝑛) from Lemma 4: 𝐾𝑓𝐿𝑇_𝑚×𝑛=𝑟2𝐾𝑓𝑇_𝑚×𝑛+(𝑟−2)𝑚²𝑛²8=2𝐾𝑓𝑇_𝑚×𝑛+𝑚²𝑛²4. (15)
Substituting the results of Lemma 7 into (15), we can get the formula for the Kirchhoff index of 𝐾𝑓(𝐿(𝑇_𝑚×𝑛)), 𝐾𝑓𝐿𝑇_𝑚×𝑛=2𝑚𝑛^𝑚−1_𝑖=1^𝑛−1_𝑗=114sin²(𝑖𝜋/𝑚)+4sin²𝑚𝑗𝜋/𝑛+𝑛³−𝑚6𝑛+𝑚³−𝑛6+𝑚²𝑛²4, (16) which completes the proof.

3.2. The Kirchhoff Index of 𝑆(𝑇_𝑚×𝑛)

In an almost identical way as Theorem 9, we derived the formula for the Kirchhoff index on the subdivision graph of 𝑇_𝑚×𝑛, denoted by Kf(𝑆(𝑇_𝑚×𝑛)).

Theorem 10. Let 𝑆(𝑇_𝑚×𝑛) be subdivision graphs of 𝑇_𝑚×𝑛 with any positive integer 𝑚,𝑛≥3; then 𝐾𝑓𝑆𝑇_𝑚×𝑛=18𝑚𝑛^𝑚−1_𝑖=1^𝑛−1_𝑗=114sin²(𝑖𝜋/𝑚)+4sin²𝑚𝑗𝜋/𝑛+3𝑛³−𝑚2𝑛+3𝑚³−𝑛2+3𝑚²𝑛²+𝑚𝑛2. (17)

Proof. Noting that 𝑇_𝑚×𝑛 are 4-regular graphs which have 𝑚𝑛 vertices, we clearly obtained from Lemma 5𝐾𝑓𝑆𝑇_𝑚×𝑛=(𝑟+2)²2𝐾𝑓𝑇_𝑚×𝑛+𝑟²−4𝑚²𝑛²+4𝑚𝑛8=18𝐾𝑓𝑇_𝑚×𝑛+3𝑚²𝑛²+𝑚𝑛2. (18)
Together with the results of Lemma 7 and (18), we can get the formula for the Kirchhoff index on the subdivision graph of 𝑇_𝑚×𝑛: 𝐾𝑓𝑆𝑇_𝑚×𝑛=18𝑚𝑛^𝑚−1_𝑖=1^𝑛−1_𝑗=114sin²(𝑖𝜋/𝑚)+4sin²𝑚𝑗𝜋/𝑛+3𝑛³−𝑚2𝑛+3𝑚³−𝑛2+3𝑚²𝑛²+𝑚𝑛2. (19) The proof is completed.

3.3. The Kirchhoff Index of 𝑇(𝑇_𝑚×𝑛)

Now we proved the formula for estimating the Kirchhoff index in the total graph of 𝑇_𝑚×𝑛, denoted by Kf(𝑇(𝑇_𝑚×𝑛)).

Theorem 11. Let 𝑇(𝑇_𝑚×𝑛) be total graphs of 𝑇_𝑚×𝑛 with any positive integer 𝑚,𝑛≥3; then 𝐾𝑓𝑇𝑇_𝑚×𝑛=187𝑚𝑛^𝑚−1_𝑖=1^𝑛−1_𝑗=114sin²(𝑖𝜋/𝑚)+4sin²+𝑗𝜋/𝑛24𝑚𝑛7^𝑚−1_𝑖=0^𝑛−1_𝑗=017+4sin²(𝑖𝜋/𝑚)+4sin²+3𝑗𝜋/𝑛𝑛𝑚14³+3−𝑚𝑚𝑛14³−𝑛+3𝑚²𝑛²+10𝑚𝑛2,𝑖,𝑗≠(0,0). (20)

Proof. Supposing that the Laplacian eigenvalues of 𝐿_{(𝐶𝑚□𝐶𝑛)} are 𝜇_𝑖𝑗, one can readily see that 𝜇_𝑖𝑗=4sin²𝑖𝜋𝑚+4sin²𝑗𝜋𝑛,𝑖=0,1,…,𝑚−1;𝑗=0,1,…,𝑛−1. (21)
Applying Lemma 6, the following result is straightforward: 𝐾𝑓𝑇𝑇_𝑚×𝑛=187𝐾𝑓𝑇_𝑚×𝑛+3𝑚²𝑛²+10𝑚𝑛2+24𝑚𝑛7_{(𝑖,𝑗)≠(0,0)}17+𝜇_𝑖𝑗. (22)
Notice that 𝐿_{(𝐶𝑚□𝐶𝑛)} have 𝑚𝑛−1 nonzero Laplacian eigenvalues, and _{(𝑖,𝑗)≠(0,0)}17+𝜇_𝑖𝑗=^𝑚−1_𝑖=0^𝑛−1_𝑗=017+4sin²(𝑖𝜋/𝑚)+4sin²,≠(),𝑗𝜋/𝑛𝑖,𝑗0,0 (23) where 𝑖=0,1,…,𝑚−1 and 𝑗=0,1,…,𝑛−1.
Consequently, the relationships between 𝑇_𝑚×𝑛 and its variant networks 𝑇(𝑇_𝑚×𝑛) for Kirchhoff index are as follows: 𝐾𝑓𝑇𝑇_𝑚×𝑛=187𝐾𝑓𝑇_𝑚×𝑛+24𝑚𝑛7^𝑚−1_𝑖=0^𝑛−1_𝑗=017+4sin²(𝑖𝜋/𝑚)+4sin²+𝑗𝜋/𝑛3𝑚²𝑛²+10𝑚𝑛2,𝑖,𝑗≠(0,0). (24)
According to the results of Lemma 7, we can verify the formula for the Kirchhoff index of the total graph of 𝐾𝑓(𝑇(𝑇_𝑚×𝑛)) from (24). Consider 𝐾𝑓𝑇𝑇_𝑚×𝑛=187𝑚𝑛^𝑚−1_𝑖=1^𝑛−1_𝑗=114sin²(𝑖𝜋/𝑚)+4sin²+𝑗𝜋/𝑛24𝑚𝑛7^𝑚−1_𝑖=0^𝑛−1_𝑗=017+4sin²(𝑖𝜋/𝑚)+4sin²+3𝑗𝜋/𝑛𝑛𝑚14³+3−𝑚𝑚𝑛14³+−𝑛3𝑚²𝑛²+10𝑚𝑛2,𝑖,𝑗≠(0,0). (25) This completes the proof of Theorem 11.

3.4. The Kirchhoff Index of 𝐶(𝑇_𝑚×𝑛)

We will explore the formula for estimating the Kirchhoff index in the clique-inserted graph of 𝑇_𝑚×𝑛, denoted by 𝐾𝑓(𝐶(𝑇_𝑚×𝑛)).

Theorem 12. Let 𝐶(𝑇_𝑚×𝑛) be clique-inserted graphs of 𝑇_𝑚×𝑛 with any positive integer 𝑚,𝑛≥3; then 𝐾𝑓𝐶𝑇_𝑚×𝑛=4𝑚𝑛^𝑚−1_𝑖=0^𝑛−1_𝑗=013−5+2cos(2𝜋𝑖/𝑚)+2cos2𝜋𝑗/𝑛+4𝑚𝑛^𝑚−1_𝑖=0^𝑛−1_𝑗=013+()+55+2cos2𝜋𝑖/𝑚+2cos2𝜋𝑗/𝑛3𝑚²𝑛², (26) where the first summation 𝑖=0,1,…,𝑚−1; 𝑗=0,1,…,𝑛−1, and (𝑖,𝑗)≠(0,0).

Proof. Noting that 𝑆(𝑇_𝑚×𝑛) is (𝑟,2)-semiregular graphs and supposing that 𝑆(𝑇_𝑚×𝑛)has  𝑝 vertices and 𝑞 edges, then obviously 𝑝=3𝑚𝑛, 𝑞=4𝑚𝑛, and 𝑟=4, respectively.
By virtue of Lemma 3, 𝑃_𝐿(𝐺)(𝑥)=(−1)^𝑛(𝑥−(𝑟+𝑠))^𝑚−𝑛𝑃_𝐺(𝑟+𝑠−𝑥). (27) Let 𝐺 be the graph 𝑆(𝑇_𝑚×𝑛); that is, 𝑃_{𝐿(𝑆(𝑇𝑚×𝑛))}(𝑥)=(−1)^𝑝(𝑥−(𝑟+𝑠))^𝑞−𝑝𝑃_{𝑆(𝑇𝑚×𝑛)}(𝑟+𝑠−𝑥). (28) From the definition of clique-inserted graph, one can immediately obtain that 𝑃_{𝐶(𝑇𝑚×𝑛)}(𝑥)=(−1)^𝑝(𝑥−(𝑟+2))^𝑞−𝑝𝑃_{𝑆(𝑇𝑚×𝑛)}()=()6−𝑥−1^3𝑚𝑛(𝑥−6)^𝑚𝑛𝑃_{𝑆(𝑇𝑚×𝑛)}(6−𝑥). (29)
Obviously, it follows from Lemma 2, 𝑃_𝑆(𝐺)(𝑥)=(−1)^𝑚(2−𝑥)^𝑚−𝑛𝑃_𝐺(𝑥(𝑟+2−𝑥)). (30) Replace 𝑥 with 6−𝑥 in (30); moreover, 𝑇_𝑚×𝑛 have 𝑚𝑛 vertices and 2𝑚𝑛 edges; we have that 𝑃_{𝑆(𝑇𝑚×𝑛)}(6−𝑥)=(−1)^𝑚𝑛(𝑥−4)^𝑚𝑛𝑃_{𝑇𝑚×𝑛}(𝑥(6−𝑥)). (31)
Based on (29) and (31), 𝑃_{𝐶(𝑇𝑚×𝑛)}(𝑥)=(−1)^4𝑚𝑛((𝑥−4)(𝑥−6))^𝑚𝑛𝑃_{𝑇𝑚×𝑛}(𝑥(6−𝑥)). (32) Since the roots of 𝑥(6−𝑥)=𝜇_𝑖𝑗 are 3±9−𝜇_𝑖𝑗, (33) where 𝜇_𝑖𝑗 are the Laplacian eigenvalues of 𝑇_𝑚×𝑛 and 𝜇_𝑖𝑗=4sin²(𝑖𝜋/𝑚)+4sin²(𝑗𝜋/𝑛), 𝑖=0,1,…,𝑚−1; 𝑗=0,1,…,𝑛−1.
It follows from (32) that the Laplacian spectrum of 𝐶(𝑇_𝑚×𝑛) is Spec_𝐿𝐶𝑇_𝑚×𝑛=46𝜇_𝑖𝑗𝐶𝑇_𝑚×𝑛𝜇^′_𝑖𝑗𝐶𝑇_𝑚×𝑛,𝑚𝑛𝑚𝑛11 (34) where 𝜇_𝑖𝑗(𝐶(𝑇_𝑚×𝑛))=3−9−𝜇_𝑖𝑗, 𝜇^′_𝑖𝑗(𝐶(𝑇_𝑚×𝑛))=3+9−𝜇_𝑖𝑗, 𝑖=0,1,…,𝑚−1, and 𝑗=0,1,…,𝑛−1.
Employing Lemma 1, (33), and the Laplacian spectrum of 𝐶(𝑇_𝑚×𝑛), the following result is straightforward: 𝐾𝑓𝐶𝑇_𝑚×𝑛=4𝑚𝑛𝑚𝑛4+𝑚𝑛6+_{≠()𝑖,𝑗0,0}1𝜇_𝑖𝑗(𝐶(𝐺1))+𝜇^′_𝑖𝑗(𝐶(𝐺=5))3𝑚²𝑛²+4𝑚𝑛_{≠()𝑖,𝑗0,0}13−9−𝜇_𝑖𝑗1+3+9−𝜇_𝑖𝑗=4𝑚𝑛^𝑚−1_𝑖=0^𝑛−1_𝑗=013−(),≠(),5+2cos2𝜋𝑖/𝑚+2cos2𝜋𝑗/𝑛𝑖,𝑗0,0+4𝑚𝑛^𝑚−1_𝑖=0^𝑛−1_𝑗=013+()+55+2cos2𝜋𝑖/𝑚+2cos2𝜋𝑗/𝑛3𝑚²𝑛². (35) Hence Theorem 12 holds.

Remark 13. The consequences of Lemma 7 and Theorems 9–12 above present closed-form formulae for immediately obtaining its Kirchhoff indexes in terms of finite various networks; however, the quantities are rather difficult to calculate directly.

4. The Asymptotic Behavior of Related Kirchhoff Index

We explore the asymptotic behavior of Kirchhoff index for the investigated networks above as 𝑚,𝑛 tend to infinity. It is interesting and surprising that the quantity tends to a constant even though Kf(𝐺)→∞, as 𝑚,𝑛 tend to infinity; that is, lim_𝑚→∞lim_𝑛→∞𝐾𝑓(𝐺)𝑚²𝑛²=𝐶,𝑚,𝑛⟶∞. (36) Moreover, one can employ the applications of analysis approach to obtain the explicit approximate values of Kirchhoff index for the related networks.

Theorem 14. Let 𝐿(𝑇_𝑚×𝑛) be line graphs of 𝑇_𝑚×𝑛 with any positive integer 𝑚,𝑛; then lim_𝑚→∞lim_𝑛→∞𝐾𝑓𝐿𝑇_𝑚×𝑛𝑚²𝑛²≈4.060. (37)

Proof. According to (15) and the result of Lemma 8, we can derive that 𝐾𝑓𝐿𝑇_𝑚×𝑛=2𝐾𝑓𝑇_𝑚×𝑛+𝑚²𝑛²4=4.060𝑚²𝑛². (38) Consequently, lim_𝑚→∞lim_𝑛→∞𝐾𝑓𝐿𝑇_𝑚×𝑛𝑚²𝑛²≈4.060. (39)
The result is equivalent to 𝐿(𝑇_𝑚×𝑛) having asymptotic Kirchhoff index, 𝐾𝑓𝐿𝑇_𝑚×𝑛≈4.060𝑚²𝑛²,𝑚,𝑛⟶∞. (40)

Theorem 15. Let 𝑆(𝑇_𝑚×𝑛) be subdivision graph of 𝑇_𝑚×𝑛 with any positive integer 𝑚,𝑛; then lim_𝑚→∞lim_𝑛→∞𝐾𝑓𝑆𝑇_𝑚×𝑛𝑚²𝑛²≈35.790. (41)

Proof. Similarly, according to (18), we can easily verify that 𝐾𝑓𝑆𝑇_𝑚×𝑛=18𝐾𝑓𝑇_𝑚×𝑛+3𝑚²𝑛²+𝑚𝑛2=35.790𝑚²𝑛². (42) Hence, lim_𝑚→∞lim_𝑛→∞𝐾𝑓𝑆𝑇_𝑚×𝑛𝑚²𝑛²≈35.790. (43)

Theorem 16. Let 𝑇(𝑇_𝑚×𝑛) be total graph of 𝑇_𝑚×𝑛 with any positive integer 𝑚,𝑛≥3; then lim_𝑚→∞lim_𝑛→∞𝐾𝑓𝑇𝑇_𝑚×𝑛𝑚²𝑛²≈5.521. (44)

Proof. Consider the summation term ∑^𝑚−1_𝑖=0∑^𝑛−1_𝑗=0(1/(7+4sin²(𝑖𝜋/𝑚)+4sin²(𝑗𝜋/𝑛))).
Since lim_𝑚→∞lim_𝑛→∞1𝑚1𝑛^𝑚−1_𝑖=0^𝑛−1_𝑗=017+4sin²(𝑖𝜋/𝑚)+4sin²𝑗𝜋/𝑛=lim_𝑚→∞lim_𝑛→∞1𝑚1𝑛^𝑚−1_𝑖=0^𝑛−1_𝑗=01()=111−2cos2𝜋𝑖/𝑚−2cos2𝜋𝑗/𝑛4𝜋²₀^2𝜋₀^2𝜋𝑑_𝑥𝑑_𝑦11−2cos𝑥−2cos𝑦≈0.094. (45) The value in last line via the mathematic software MATLAB, which can obtain the result above.
Combining with (22), we can obtain that 𝐾𝑓𝑇𝑇_𝑚×𝑛=187𝐾𝑓𝑇_𝑚×𝑛+24𝑚𝑛7^𝑚−1_𝑖=0^𝑛−1_𝑗=017+4sin²(𝑖𝜋/𝑚)+4sin²+𝑗𝜋/𝑛3𝑚²𝑛²+10𝑚𝑛2,𝑖,𝑗≠(0,0)≈5.521𝑚²𝑛². (46) So lim_𝑚→∞lim_𝑛→∞𝐾𝑓𝑇𝑇_𝑚×𝑛𝑚²𝑛²≈5.521. (47)

Theorem 17. Let 𝐶(𝑇_𝑚×𝑛) be clique-inserted graph of 𝑇_𝑚×𝑛 with any positive integer 𝑚,𝑛≥3; then lim_𝑚→∞lim_𝑛→∞𝐾𝑓𝐶𝑇_𝑚×𝑛𝑚²𝑛²≈38.591. (48)

Proof. From the proof of Theorem 12, we know that 𝐾𝑓𝐶𝑇_𝑚×𝑛=4𝑚𝑛^𝑚−1_𝑖=0^𝑛−1_𝑗=013−5+2cos(2𝜋𝑖/𝑚)+2cos2𝜋𝑗/𝑛+4𝑚𝑛^𝑚−1_𝑖=0^𝑛−1_𝑗=013+()+55+2cos2𝜋𝑖/𝑚+2cos2𝜋𝑗/𝑛3𝑚²𝑛², (49) where the first summation 𝑖=0,1,…,𝑚−1; 𝑗=0,1,…,𝑛−1, and (𝑖,𝑗)≠(0,0).
As 𝑚,𝑛 tend to infinity, it follows from the first summation term: lim_𝑚→∞lim_𝑛→∞1𝑚1𝑛^𝑚−1_𝑖=0^𝑛−1_𝑗=013−()=15+2cos2𝜋𝑖/𝑚+2cos2𝜋𝑗/𝑛4𝜋²₀^2𝜋₀^2𝜋𝑑_𝑥𝑑_𝑦3−5+2cos𝑥+2cos𝑦≈9.037. (50) Similarly, it holds from the second summation term when 𝑚,𝑛 tend to infinity, lim_𝑚→∞lim_𝑛→∞1𝑚1𝑛^𝑚−1_𝑖=0^𝑛−1_𝑗=013+()=15+2cos2𝜋𝑖/𝑚+2cos2𝜋𝑗/𝑛4𝜋²₀^2𝜋₀^2𝜋𝑑_𝑥𝑑_𝑦3+5+2cos𝑥+2cos𝑦≈0.195. (51) Combining with the consequences of Theorem 12 and (50) and (51), it follows that lim_𝑚→∞lim_𝑛→∞𝐾𝑓𝐶𝑇_𝑚×𝑛𝑚²𝑛²≈38.591. (52) Summing up, we complete the proof.

5. Conclusions

Resistance distance was introduced by Klein and Randić as a generalization of the classical distance. In this paper, we have deduced the relationships between the toroidal meshes network 𝑇_𝑚×𝑛 and its variant networks in terms of the Kirchhoff index via spectral graph theory. The explicit formulae for calculating the Kirchhoff indexes of 𝐿(𝑇_𝑚×𝑛), 𝑆(𝑇_𝑚×𝑛), 𝑇(𝑇_𝑚×𝑛), and 𝐶(𝑇_𝑚×𝑛) were proposed for any positive integer 𝑚,𝑛≥3, respectively.

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.

Conflict of Interests

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

Acknowledgments

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.