The Kirchhoff Index of Toroidal Meshes and Variant Networks

1 School of Mathematical Sciences, Anhui University, Hefei 230601, China 2Department of Mathematics, Southeast University, Nanjing 210096, China 3Department of Public Courses, Anhui Xinhua University, Hefei 230088, China 4Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 5 College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China

Given graphs  and  with vertex sets  and , the Cartesian product  ◻  of graphs  and  is a graph such that the vertex set of  ◻  is the Cartesian product  × ; and any two vertices (,   ) and (V, V  ) are adjacent in  ◻  if and only if either  = V and   is adjacent with V  in  or   = V  and  is adjacent with V in  [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 (), is defined as a line graph of the subdivision graph () [2,3].The subdivision graph of an -regular graph is (, 2)semiregular graph.Consequently, the clique-inserted graph of an -regular graph is the line graph of an (, 2)-semiregular graph.
The resistance distances between vertices  and , denoted by   , are defined as the effective electrical resistance between them if each edge of  is replaced by a unit resistor [4].A famous distance-based topological index, the Kirchhoff index Kf(), is defined as the sum of resistance distances between all pairs of vertices in ; that is, Kf() = (1/2) ∑  =1 ∑  =1   (), known as the Kirchhoff index of  [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][7][8][9][10][11][12][13].Besides, the Kirchhoff index also is a structure descriptor [14].Unfortunately, it is rather hard to directly design some algorithms [15][16][17] to calculate resistance distances and the Kirchhoff indexes of graphs.So, many researchers investigated some special classes of graphs [18][19][20][21].In addition, many efforts were also made to obtain the Kirchhoff index bounds for some graphs [17,22].Details on 2 Mathematical Problems in Engineering 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 ( × ), ( × ), ( × ), and ( × ) 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.

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  × stands for the graphs   ◻   for the convenience of description.It is trivial for ,  are 1, 2, without loss of generality, we discuss the situations for any positive integer ,  ≥ 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  be a connected graph with  ≥ 2 vertices and let  1 ≥  2 ≥ ⋅ ⋅ ⋅ ≥   = 0 be the Laplacian eigenvalues of graph ; then The line graph of a graph , denoted by (), is the graph whose vertices correspond to the edges of  with two vertices of () being adjacent if and only if the corresponding edges in  share a common vertex.The subdivision graph of a graph , denoted by (), is the graph obtained by replacing every edge in  with a copy of  2 (path of length two).The total graph of a graph , denoted by (), is the graph whose vertices correspond to the union of the set of vertices and edges of , with two vertices of () being adjacent if and only if the corresponding elements are adjacent or incident in .Let   () be the characteristic polynomial of the Laplacian matrix of a graph ; the following results were shown in [23].
A bipartite graph  with a bipartition () = (, ) is called an (, )-semiregular graph if all vertices in  have degree  and all vertices in  have degree .Apparently, the subdivision graph of an -regular-graph  is (, 2)semiregular graph.
Lemma 3 (see [24]).Let  be an (, )-semiregular connected graph with  vertices.Then where  () () is the Laplacian characteristic polynomial of the line graph () and  is the number of edges of .
Lemma 4 (see [23]).Let  be a connected simple r-regular graph with  vertices and  edges and let () be the line graph of .Then Lemma 5 (see [23]).Let  be a connected simple -regular graph with  ≥ 2 vertices; then The following lemma gives an expression on Kf (()) and Kf () of a regular graph .Lemma 6 (see [25]).Let  be a -regular connected graph with  vertices and  edges, and  ≥ 2; then Kf ( ()) =  ( + 2) ( + 4) 2 ( + 3) Lemma 7 presents the formula for calculating Kirchhoff index of  × ; in the following proof, some techniques in [26] are referred to.
The following consequence was presented in [26].Here we give a short proof.Lemma 8 (see [26]).For the toroidal networks  × with any positive integer ,  ≥ 3, Proof.By virtue of ( 9), one can derive that Hence, lim

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  × , denoted by Kf(( × )).
Theorem 9. Let ( × ) be line graphs of  × with any positive integer ,  ≥ 3; then Proof.Apparently the toroidal networks  × are 4-regular graphs which have  vertices and 2 edges, respectively.We clearly obtained the following relationship Kf (( × )) and Kf ( × ) from Lemma 4: Substituting the results of Lemma 7 into (15), we can get the formula for the Kirchhoff index of Kf (( × )), which completes the proof.

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  × , denoted by Kf(( × )).
Theorem 10.Let ( × ) be subdivision graphs of  × with any positive integer ,  ≥ 3; then Proof.Noting that  × are 4-regular graphs which have  vertices, we clearly obtained from Lemma 5 Together with the results of Lemma 7 and ( 18), we can get the formula for the Kirchhoff index on the subdivision graph of  × : The proof is completed.

The Kirchhoff Index of 𝑇(𝑇 𝑚×𝑛
).Now we proved the formula for estimating the Kirchhoff index in the total graph of  × , denoted by Kf(( × )).
Consequently, the relationships between  × and its variant networks ( × ) for Kirchhoff index are as follows: According to the results of Lemma 7, we can verify the formula for the Kirchhoff index of the total graph of Kf (( × )) from (24) This completes the proof of Theorem 11.

The Kirchhoff Index of 𝐶(𝑇 𝑚×𝑛
).We will explore the formula for estimating the Kirchhoff index in the clique-inserted graph of  × , denoted by Kf (( × )).
Employing Lemma 1, (33), and the Laplacian spectrum of ( × ), the following result is straightforward: 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.

The Asymptotic Behavior of Related Kirchhoff Index
We explore the asymptotic behavior of Kirchhoff index for the investigated networks above as ,  tend to infinity.It is interesting and surprising that the quantity tends to a constant even though Kf() → ∞, as ,  tend to infinity; that is, lim