#### Abstract

The number of spanning trees in graphs or in networks is an important issue. The evaluation of this number not only is interesting from a mathematical (computational) perspective but also is an important measure of reliability of a network or designing electrical circuits. In this paper, a simple formula for the number of spanning trees of the Cartesian product of two regular graphs is investigated. Using this formula, the number of spanning trees of the four well-known regular networks can be simply taken into evaluation.

#### 1. Introduction

In this paper, we deal with simple undirected graphs having no self-loop or multiple edges and consider the Cartesian product of two regular graphs only. It is well known that, for designing large-scale interconnection networks, the Cartesian product is an important method to obtain large networks from smaller ones, with a number of parameters that can be easily calculated from the corresponding parameters for those small initial graphs. The Cartesian product preserves many nice properties such as regularity, transitivity, super edge-connectivity, and super point-connectivity of the initial regular graphs [1–6]. In fact, many well-known networks can be constructed by the Cartesian products of simple regular graphs, for example, Boolean -cube networks, hypercube networks, and lattice networks.

Alternatively, the study of the number of spanning trees in a graph has a long history and has been very active because the problem has different practical applications in different fields. For example, the number characterizes the reliability of a network and, in physics, designing electrical circuits, analyzing energy of masers, and investigating the possible particle transitions [7–10]. The larger degree of points a network has, the more I/O ports and edges are needed and the more cost is required.

The number of spanning trees of some special network has been taken into evaluation [11–20]. Recently, some authors derived results about the counting where the number of spanning trees can be found from [21–29]. However, the study for spanning trees of the Cartesian product of regular graphs remains an open and important invariant.

The number of spanning trees of Boolean -cube networks, lattice networks, and generalized Boolean -cube networks has been taken into account [13, 17, 18]; these networks belong to the class of networks with two regular graphs and which is defined recursively by for . In this paper, we will present the formula of the number of spanning trees of the Cartesian product of regular graphs. Using this present formula, the main results in [13, 17, 18] can be obtained much more simply and will be extended.

#### 2. The Number of Spanning Trees

*Definition 1. *Let be a graph with points labeled . The adjacency matrix of , is an matrix with the th row and th column entry given by
The Kirchhoff matrix of , is equal to , where is an diagonal matrix whose diagonal entries are the degree of point and is the adjacency matrix. Thus the th row and th column entry is given by

Lemma 2 (see [30]). *If is a graph on points with Kirchhoff matrix and is the submatrix of obtained by removing the th row and th column then the number of spanning trees of , , is any cofactor of . That is, .*

Lemma 3. *If is an triangulable matrix, which has eigenvalues, then the sum of product of any eigenvalues of is the sum of all principal minors of .*

*Proof. *Let be the character polynomial of and are eigenvalues of . Then
From (3), we obtain the following.

(a) The coefficient of .

On the other hand,
where and for .

So we only need to prove that the coefficient of in is the sum of all principal minors of . Let denote the principal minor of obtained by removing the th row and th column from .

By (4), we obtain the following:

(b) So the coefficient of the coefficient of.

Hence the theorem is proved due to (a) and (b).

Since a real symmetric matrix is with the property that the sum of its rows (and its columns) is zero, the rank of . So 0 is the smallest eigenvalue. We write the eigenvalues of as an ordered list: The main result in Kelmans and Chelnokov [31] can also be obtained by the following method.

Lemma 4 (see [31]). *If the eigenvalues of the Kirchhoff matrix of the points graph are then , the number of spanning trees of , is given by
*

*Proof. *By Lemmas 2 and 3,
Hence .

Lemma 5. *Let the eigenvalues of the adjacency matrix of the regular graph be written by , where is the degree of the regular graph G; then, the number of spanning trees of is given by
*

*Proof. *We know , where is the identity matrix. Since is the eigenvalue of for , there exists eigenvector for , such that . So , , , and .

We obtain . Thus is the eigenvalue of for .

Hence the lemma is proved by Lemma 4.

#### 3. Cartesian Product and Kronecker Product

*Definition 6. *Let denote a connected graph with set of all points and set of all edges in and let denote edge joining points and . Let for ; the Cartesian product of and is defined by , where , , and if and only if and or and .

*Definition 7 (see [32]). *Let be an matrix and an matrix; then, the Kronecker product is defined as the matrix with block description
The Kronecker sum is defined by , where is the identity matrix for . Let be an matrix. can be partitioned into blocks which are denoted by for and . That is,
where is the matrix for and . is called an block matrix.

Lemma 8. *If is an matrix and , are matrices then*(1)*,*(2)*.*

*Proof. *The lemma is easily obtained.

Lemma 9. *If the products and are defined then .*

*Proof. *Let be an matrix and an matrix

Lemma 10. *If and are invertible then .*

*Proof. *Consider
where is and is .

#### 4. The Number of Spanning Trees of the Cartesian Product of Regular Graphs

Lemma 11. *If the points of and are labeled by and , respectively, and points of are ordered lexicographically, that is, the label of is smaller than that of if and only if and , then .*

*Proof. *Since is an matrix, is an block matrix. By the definition of , we describe

where is the entry of the adjacent matrix of and is the adjacent matrix of . We know is an matrix; it can be described as an block matrix

where is the zero matrix. Since is an matrix, it can be described as an block matrix

Clearly .

Lemma 12. *Let be the regular graph of degree for ; then, the degree of is . If the number of the points of (resp., ) is (resp., ) and the points of are ordered lexicographically then .*

*Proof. *By Lemmas 8 and 11,
where is the identity matrix.

Lemma 13. *If and are triangulable matrices then the eigenvalues of are given by , respectively, as and vary through the eigenvalues of and .*

*Proof. *Since and are triangulable, there exist invertible matrices and such that and are upper triangular. If and are and matrices, respectively, by Lemmas 9 and 10,
So is similar to and they have the same eigenvalues. Obviously is upper triangular with diagonal entries given by , respectively, as and vary through the eigenvalues of and . Hence the eigenvalues of are , respectively, as and vary through the eigenvalues of and .

Theorem 14. *Let and be the regular graphs with degrees and , respectively. If the eigenvalues of the adjacency matrix are written as and the eigenvalues of the adjacency matrix are written as , then the number of spanning trees of the Cartesian product is
**
where and satisfy .*

*Proof. *We know has points and the degree of is . By Lemma 11, . By Lemma 13, the eigenvalues of are for and . The result follows by Lemma 5.

Theorem 15. *If and are the regular graph of degrees and , respectively, the eigenvalues of the Kirchhoff matrix are written as , and the eigenvalues of the Kirchhoff matrix are written as , then the number of spanning trees of the Cartesian product of and is
**
where and satisfy .*

*Proof. *By Lemma 12, . By Lemma 13, the eigenvalues of are for and . Hence by Lemma 4, the result follows.

#### 5. The Number of Spanning Trees of the -Lattice Network

*Definition 16 (see [17, 18]). *The -lattice networks are defined as
where is a complete graph of points.

When , is well known, the Boolean -cube network.

We denote .

Lemma 17. *The eigenvalues of are with multiplicity and 0 with multiplicity 1.*

*Proof. *Since , where is the matrix of all ones, letting be the character polynomial of , we obtain by Gaussian elimination
Hence the result follows.

Lemma 18. *If the distinct eigenvalues of the Kirchhoff matrix are then**
where .*

*Proof. *Since and by Lemma 13, we obtain each of eigenvalues of one of eigenvalues of .

Hence if we take the eigenvalue 0 of for then with multiplicity . If we take the eigenvalue of for some one and the eigenvalue 0 of for each then with multiplicity . If we take the eigenvalue of and , respectively, for and the eigenvalue 0 of for each , then with multiplicity . We keep performing the same process. Hence the result follows.

The main theorem in [17, 18] can be obtained much more simply by Theorem 19 as follows.

Theorem 19 (see [17]). *The number of spanning trees of is
*

*Proof. *Since the degree of is , the number of points of is . By Lemma 18 and Theorem 15, we obtain

Corollary 20 (see [18]). *The number of spanning trees of the Boolean -cube network is
*

*Proof. *Since , by Theorem 15, the result follows.

#### 6. The Number of Spanning Trees of the -Lattice Network

*Definition 21. *The lattice network can be defined recursively by and .

Thus has points. We denote .

Theorem 22. *The number of spanning trees of is
*

*Proof. *Since the eigenvalues of are with multiplicity and 0 with multiplicity 1, the distinct eigenvalues of are 0 and with multiplicity , where , satisfying , , are nonzero eigenvalues of , respectively, and take zero eigenvalues for the remaining , where , . By Lemma 13 and Theorem 15, the result follows.

*Example 23. *The number of spanning trees of and is as shown in Figure 1, where and = 1620609272381440.

#### 7. The Number of Spanning Trees of the Generalized Boolean -Cube Network

*Definition 24. *The generalized Boolean -cube network can be defined by
where is a cycle with points. One denotes .

Setting is the matrix by

Lemma 25. *The eigenvalues of the adjacent matrix are with multiplicity and with multiplicity 1.*

Lemma 26 (see [33]). *If is a sequence matrix, is an eigenvalue of , and is a polynomial then is the eigenvalue of .*

Lemma 27. *The eigenvalues of the adjacent matrix are for .*

*Proof. *Since
the eigenvalues of are for . It follows that . By Lemma 26, the eigenvalues of are for .

The main theorem in [13] can be obtained much more simply as follows.

Theorem 28 (see [13]). *The number of spanning trees of is
*

*Proof. *It follows that the points of are and the degree of any edge of is . By Lemma 25, the eigenvalues of the adjacent matrix are and 1. By Lemmas 13 and 27, the distinct eigenvalues of the adjacent matrix are
where .

When and , the eigenvalue is . When and , the eigenvalues are . By Theorem 14,
Since = and , hence
Since and as ,

#### 8. The Number of Spanning Trees of the Hypercube Network

*Definition 29. *The hypercube network can be defined by
where is a cycle with points.

Theorem 30. *The number of spanning trees of is
**
where satisfy .*

*Proof. *It follows that the points of are and the degree of any edge of is . By Lemma 26 and Theorem 14,
where satisfy .

#### 9. Conclusion

Due to the high dependence of the network design and reliability problem, electrical circuits designing issue are on the graph theory. For example, the larger degree of points a network has, the more I/O ports and edges are needed and the more cost is required. The evaluation of this number not only is interesting from a mathematical (computational) perspective but also is an important issue on practical applications. However, the study for spanning trees of the Cartesian product of regular graphs remains an open and important invariant. In this paper, the eigenvalues of the Kirchhoff matrix of Cartesian product of two regular graphs, and , are given by as and vary through the eigenvalues of the Kirchhoff matrices and , respectively. By this result, the formula for the number of spanning trees of the four regular networks can be simply obtained. Using this formula, the main results in [13, 17, 18] can be obtained much more simply and will be extended.

#### Conflict of Interests

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

#### Acknowledgment

The authors would like to thank the anonymous reviewers for their valuable suggestions.