The Number of Spanning Trees of the Cartesian Product of Regular Graphs

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 𝑛-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 𝑛-cube networks, lattice networks, and generalized Boolean 𝑛-cube networks has been taken into account [13, 17, 18]; these networks belong to the class of networks 𝑄_𝑛 with two regular graphs 𝑄₁ and 𝐺 which is defined recursively by 𝑄_𝑛=𝑄_𝑛−1×𝐺 for 𝑛≥2. 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 𝐺 be a graph with 𝑛 points labeled 1,2,…,𝑛. The adjacency matrix of 𝐺,𝐴(𝐺), is an 𝑛×𝑛 matrix with the 𝑖th row and 𝑗th column entry given by[𝐴(𝐺)]_𝑖𝑗=1ifpoints𝑖and𝑗areadjacent0ortherwise. (1) The Kirchhoff matrix of 𝐺,𝐻(𝐺), is equal to 𝐷(𝐺)−𝐴(𝐺), where 𝐷(𝐺) is an 𝑛×𝑛 diagonal matrix whose diagonal entries are the degree of point 𝑛 and 𝐴(𝐺) is the adjacency matrix. Thus the 𝑖th row and 𝑗th column entry is given by [𝐻(𝐺)]_𝑖𝑗=deg(𝑖)if𝑖=𝑗−1if𝑖≠𝑗andpoints𝑖and𝑗areadjacent0ortherwise. (2)

Lemma 2 (see [30]). If 𝐺 is a graph on 𝑛 points with Kirchhoff matrix 𝐻(𝐺) and 𝐻_𝑖𝑗(𝐺) is the submatrix of 𝐻(𝐺) obtained by removing the 𝑖th row and 𝑗th column then the number of spanning trees of 𝐺,  𝑡(𝐺), is any cofactor of 𝐻(𝐺). That is, 𝑡(𝐺)=(−1)^𝑖+𝑗det(𝐻_𝑖𝑗(𝐺)).

Lemma 3. If 𝐴 is an 𝑛×𝑛 triangulable matrix, which has 𝑛 eigenvalues, then the sum of product of any 𝑛−1 eigenvalues of 𝐴 is the sum of all principal minors of 𝐴.

Proof. Let 𝑝(𝜆) be the character polynomial of 𝐴 and 𝜆₁,𝜆₂,…,𝜆_𝑛 are 𝑛 eigenvalues of 𝐴. Then 𝑝(𝜆)=𝜆−𝜆₁𝜆−𝜆₂⋯𝜆−𝜆_𝑛. (3) From (3), we obtain the following.
(a) The coefficient of 𝜆=(−1)^𝑛−1⋅(thesumofproductofany𝑛−1eigenvaulesof𝐴).
On the other hand, 𝑝(𝜆)()=𝜆=det𝜆𝐼−𝐴₍₁₎−𝑎₁₁−𝑎₁₂⋯−𝑎_1𝑛−𝑎₂₁𝜆⁽²⁾−𝑎₂₂⋯−𝑎_2𝑛⋮⋮⋱⋮−𝑎₂₁−𝑎₂₁⋯𝜆^(𝑛)−𝑎_𝑛𝑛, (4) where 𝐴=[𝑎_𝑖𝑗] and 𝜆^(𝑖)=𝜆 for 𝑖=1,2,…,𝑛.
So we only need to prove that the coefficient of 𝜆 in det(𝜆𝐼−𝐴) is the sum of all principal minors of 𝐴. Let 𝐴_𝑖𝑖 denote the principal minor of 𝐴 obtained by removing the 𝑖th row and 𝑖th column from 𝐴.
By (4), we obtain the following:thecoefficientof𝜆⁽¹⁾=(−1)^𝑛−1𝐴₁₁taking𝜆⁽²⁾=𝜆⁽³⁾=⋯=𝜆^(𝑛)=0thecoefficientof𝜆⁽²⁾=(−1)^𝑛−1𝐴₂₂taking𝜆⁽¹⁾=𝜆⁽³⁾=𝜆⁽⁴⁾⋯=𝜆^(𝑛)⋮=0thecoefficientof𝜆^(𝑛)=(−1)^𝑛−1𝐴_𝑛𝑛taking𝜆⁽¹⁾=𝜆⁽²⁾=⋯=𝜆^(𝑛−1)=0. (5)
(b) So the coefficient of 𝜆=∑^𝑛_𝑖=1  the coefficient of𝜆^(𝑖)=(−1)^𝑛−1(𝐴₁₁+𝐴₂₂+⋯+𝐴_𝑛𝑛).
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 𝐻(𝐺)≤𝑛−1. So 0 is the smallest eigenvalue. We write the eigenvalues of 𝐻(𝐺) as an ordered list: 0=𝜆₀(𝐺)≤𝜆₁(𝐺)≤𝜆₂(𝐺)⋯≤𝜆_𝑛−1(𝐺). (6) 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 𝐻(𝐺) of the 𝑛 points graph 𝐺 are 0=𝜆₀(𝐺)≤𝜆₁(𝐺) ≤𝜆₂(𝐺)⋯≤𝜆_𝑛−1(𝐺) then 𝑡(𝐺), the number of spanning trees of 𝐺, is given by 𝑡(𝐺)=1𝑛^𝑛−1_𝑖=1𝜆_𝑖(𝐺). (7)

Proof. By Lemmas 2 and 3, ^𝑛−1_𝑖=1𝜆_𝑖(𝐺)(𝐺)(𝐺)(𝐺).=thesumofproductofany𝑛−1eigenvaulesof𝐻=thesumofallprincipalminorsof𝐻=𝑛⋅𝑡 (8) Hence 𝑡(𝐺)=(1/𝑛)∏^𝑛−1_𝑖=1𝜆_𝑖(𝐺).

Lemma 5. Let the eigenvalues of the adjacency matrix 𝐴(𝐺) of the regular graph 𝐺 be written by 𝑢₁≤𝑢₂⋯≤𝑢_𝑛−1≤𝑢_𝑛=𝑛, where 𝑟 is the degree of the regular graph G; then, the number of spanning trees of 𝐺 is given by 𝑡(𝐺)=1𝑛^𝑛−1_𝑖=1𝑟−𝑢_𝑖. (9)

Proof. We know 𝐻(𝐺)=𝑟𝐼_𝑛−𝐴(𝐺), where 𝐼_𝑛 is the identity 𝑛×𝑛 matrix. Since 𝑢_𝑖 is the eigenvalue of 𝐴(𝐺) for 𝑖=1,2,…,𝑛−1, there exists eigenvector 𝑥_𝑖 for 𝑖=1,2,…,𝑛−1, such that 𝐴(𝐺)𝑥_𝑖=𝑢_𝑖𝑥_𝑖. So (𝑟𝐼_𝑛−𝐻(𝐺))𝑥_𝑖=𝑢_𝑖𝑥_𝑖,  𝑟𝐼_𝑛𝑥_𝑖−𝐻(𝐺)𝑥_𝑖=𝑢_𝑖𝑥_𝑖, 𝑟𝑥_𝑖−𝐻(𝐺)𝑥_𝑖=𝑢_𝑖𝑥_𝑖, and  𝐻(𝐺)𝑥_𝑖=𝑟𝑥_𝑖−𝑢_𝑖𝑥_𝑖.
We obtain 𝐻(𝐺)𝑥_𝑖=(𝑟−𝑢_𝑖)𝑥_𝑖. Thus 𝑟−𝑢_𝑖 is the eigenvalue of 𝐻(𝐺) for 𝑖=1,2,…,𝑛−1.
Hence the lemma is proved by Lemma 4.

3. Cartesian Product and Kronecker Product

Definition 6. Let 𝐺=(𝑁,𝐸) denote a connected graph with 𝑁 set of all points and 𝐸 set of all edges in 𝐺 and let {𝑢,𝑣} denote edge joining points 𝑢 and 𝑣. Let 𝐺_𝑖=(𝑁_𝑖,𝐸_𝑖) for 𝑖=1,2; the Cartesian product of 𝐺₁ and 𝐺₂ is defined by 𝐺₁×𝐺₂=(𝑁,𝐸), where 𝑁=𝑁₁×𝑁₂,  𝐸=𝐸₁×𝐸₂, and {(𝑢₁,𝑣₁),(𝑢₂,𝑣₂)}∈𝐸 if and only if 𝑢₁=𝑢₂ and {𝑣₁,𝑣₂}∈𝐸₂ or 𝑣₁=𝑣₂ and {𝑢₁,𝑢₂}∈𝐸₁.

Definition 7 (see [32]). Let 𝐵=[𝑏_𝑖𝑗] be an 𝑛×𝑛 matrix and 𝐶 an 𝑚×𝑚 matrix; then, the Kronecker product 𝐵×_𝐾𝐶 is defined as the 𝑚𝑛×𝑚𝑛 matrix with block description 𝑏₁₁𝐶𝑏₁₂𝐶⋯𝑏_1𝑛𝐶𝑏₂₁𝐶𝑏₂₂𝐶⋯𝑏_2𝑛𝐶𝑏⋮⋮⋱⋮_𝑛1𝐶𝑏_𝑛2𝐶⋯𝑏_𝑛𝑛𝐶. (10) The Kronecker sum is defined by 𝐵+_𝐾𝐶=𝐵×_𝐾𝐼_𝑚+𝐼_𝑛×_𝐾𝐶, where 𝐼_𝑘 is the 𝑘×𝑘 identity matrix for 𝑘=𝑚,𝑛. Let 𝑀 be an 𝑚𝑛×𝑚𝑛 matrix. 𝑀 can be partitioned into 𝑛² blocks which are denoted by 𝐵_𝛼𝛽 for 𝛼=1,2,…,𝑛 and 𝛽=1,2,…,𝑛. That is, 𝐵𝑀=¹¹𝐵₁₂⋯𝐵_1𝑛𝐵₂₁𝐵₂₂⋯𝐵_𝑛1𝐵⋮⋮⋱⋮_𝑛1𝐵_𝑛2⋯𝐵_𝑛𝑛, (11) where 𝐵_𝛼𝛽 is the 𝑚×𝑚 matrix for 𝛼=1,2,…,𝑛 and  𝛽=1,2,…,𝑛.  𝑀 is called an 𝑛×𝑛  (𝑚×𝑚) block matrix.

Lemma 8. If 𝐴 is an 𝑛×𝑛 matrix and 𝐵, 𝐶 are 𝑚×𝑚 matrices then(1)𝐴×_𝐾(𝐵+𝐶)=𝐴×_𝐾𝐵+𝐴×_𝐾𝐶,(2)(𝐵+𝐶)×_𝐾𝐴=𝐵×_𝐾𝐴+𝐶×_𝐾𝐴.

Proof. The lemma is easily obtained.

Lemma 9. If the products 𝐴𝐵 and 𝐶𝐷 are defined then (𝐴𝐵)×_𝐾(𝐶𝐷)=(𝐴×_𝐾𝐶)(𝐵×_𝐾𝐷).

Proof. Let 𝐴=[𝑎_𝑖𝑗] be an 𝑚×𝑛 matrix and 𝐵=[𝑏_𝑖𝑗] an 𝑛×𝑝 matrix 𝐴×_𝐾𝐶𝐵×_𝐾𝐷=𝑎₁₁𝐶⋯𝑎_1𝑛𝐶𝑎⋮⋱⋮_𝑚1𝐶⋯𝑎_𝑚𝑛𝐶𝑏₁₁𝐷⋯𝑏_1𝑝𝐷𝑏⋮⋱⋮_𝑛1𝐷⋯𝑏_𝑛𝑝𝐷=^𝑛𝑘=1𝑎_1𝑘𝑏_𝑘1𝐶𝐷⋯^𝑛_𝑘=1𝑎_1𝑘𝑏_𝑘𝑝𝐶𝐷⋮⋱⋮_𝑛𝑘=1𝑎_𝑚𝑘𝑏_𝑘1𝐶𝐷⋯^𝑛_𝑘=1𝑎_𝑚𝑘𝑏_𝑘𝑝=()×𝐶𝐷𝐴𝐵_𝐾(𝐶𝐷). (12)

Lemma 10. If 𝑃 and 𝑄 are invertible then (𝑃×_𝐾𝑄)⁻¹=𝑃⁻¹×_𝐾𝑄⁻¹.

Proof. Consider 𝑃×_𝐾𝑄𝑃⁻¹×_𝐾𝑄⁻¹=𝑃𝑃⁻¹×_𝐾𝑄𝑄⁻¹=𝐼_𝑚×_𝐾𝐼_𝑛=𝐼_𝑚𝑛, (13) where 𝑃 is 𝑚×𝑚 and 𝑄 is 𝑛×𝑛.

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

Lemma 11. If the points of 𝐺₁ and 𝐺₂ are labeled by 𝑢₁,𝑢₂,…𝑢_𝑛 and 𝑣₁,𝑣₂,…,𝑣_𝑚, respectively, and points of 𝐺₁×𝐺₂ are ordered lexicographically, that is, the label of (𝑢_𝑖,𝑣_𝑗) is smaller than that of (𝑢_𝑘,𝑣_𝑙) if and only if 𝑖<𝑘 and 𝑗<𝑙, then 𝐴(𝐺₁×𝐺₂)=𝐴(𝐺₁)+_𝐾𝐴(𝐺₂).

Proof. Since 𝐴(𝐺₁×𝐺₂) is an 𝑚𝑛×𝑚𝑛 matrix, 𝐴(𝐺₁×𝐺₂) is an 𝑛×𝑛  (𝑚×𝑚) block matrix. By the definition of 𝐺₁×𝐺₂, we describe𝐴𝐺₁×𝐺₂=𝐴𝐺₂𝐴𝐺₁₁₂𝐼_𝑚𝐴𝐺₁₁₃𝐼_𝑚⋯𝐴𝐺_11,𝑛−1𝐼_𝑚𝐴𝐺_11,𝑛𝐼_𝑚𝐴𝐺₁₂₁𝐼_𝑚𝐴𝐺₂𝐴𝐺₁₂₃𝐼_𝑚⋯𝐴𝐺_12,𝑛−1𝐼_𝑚𝐴𝐺_12,𝑛𝐼_𝑚𝐴𝐺⋮⋮⋮⋮⋮_1𝑛−1,1𝐼_𝑚𝐴𝐺_1𝑛−1,2𝐼_𝑚𝐴𝐺_1𝑛−1,3𝐼_𝑚⋯𝐴𝐺₂𝐴𝐺_{1𝑛−1,𝑛}𝐼_𝑚𝐴𝐺_1𝑛,1𝐼_𝑚𝐴𝐺_1𝑛,2𝐼_𝑚𝐴𝐺_1𝑛,3𝐼_𝑚⋯𝐴𝐺_{1𝑛,𝑛−1}𝐼_𝑚𝐴𝐺₂, (14)
where [𝐴(𝐺₁)]_𝑖𝑗 is the (𝑖,𝑗) entry of the adjacent matrix 𝐴(𝐺₁) of 𝐺₁ and 𝐴(𝐺₂) is the 𝑚×𝑚 adjacent matrix of 𝐺₂. We know 𝐼_𝑛×_𝐾𝐴(𝐺₂) is an 𝑚𝑛×𝑚𝑛 matrix; it can be described as an 𝑛×𝑛(𝑚×𝑚) block matrix𝐼_𝑛×_𝐾𝐴𝐺₂=𝐴𝐺₂0_𝑚×𝑚0_𝑚×𝑚⋯0_𝑚×𝑚0_𝑚×𝑚0_𝑚×𝑚𝐴𝐺₂0_𝑚×𝑚⋯0_𝑚×𝑚0_𝑚×𝑚0⋮⋮⋮⋯⋮⋮_𝑚×𝑚0_𝑚×𝑚0_𝑚×𝑚⋯𝐴𝐺₂0_𝑚×𝑚0_𝑚×𝑚0_𝑚×𝑚0_𝑚×𝑚⋯0_𝑚×𝑚𝐴𝐺₂, (15)
where 0_𝑚×𝑚 is the 𝑚×𝑚 zero matrix. Since 𝐴(𝐺₁)×_𝐾𝐼_𝑚 is an 𝑚𝑛×𝑚𝑛 matrix, it can be described as an 𝑛×𝑛(𝑚×𝑚) block matrix𝐴𝐺₁×_𝐾𝐼_𝑚=^{0𝑚×𝑚𝐴𝐺112𝐼𝑚𝐴𝐺113𝐼𝑚⋯𝐴𝐺11,𝑛−1𝐼𝑚𝐴𝐺11,𝑛𝐼𝑚𝐴𝐺121𝐼𝑚0𝑚×𝑚𝐴𝐺123𝐼𝑚⋯𝐴𝐺12,𝑛−1𝐼𝑚𝐴𝐺12,𝑛𝐼𝑚}_{𝐴𝐺⋮⋮⋮⋮⋮1𝑛−1,1𝐼𝑚𝐴𝐺1𝑛−1,2𝐼𝑚𝐴𝐺1𝑛−1,3𝐼𝑚⋯0𝑚×𝑚𝐴𝐺1𝑛−1,𝑛𝐼𝑚𝐴𝐺1𝑛,1𝐼𝑚𝐴𝐺1𝑛,2𝐼𝑚𝐴𝐺1𝑛,3𝐼𝑚⋯𝐴𝐺1𝑛,𝑛−1𝐼𝑚0𝑚×𝑚}. (16)
Clearly 𝐴(𝐺₁×𝐺₂)=𝐴(𝐺₁)×_𝐾𝐼_𝑚+𝐼_𝑛×_𝐾𝐴(𝐺₂)=𝐴(𝐺₁)+_𝐾𝐴(𝐺₂).

Lemma 12. Let 𝐺_𝑖 be the regular graph of degree 𝑟_𝑖 for 𝑖=1,2; then, the degree of 𝐺₁×𝐺₂ is 𝑟₁+𝑟₂. If the number of the points of 𝐺₁ (resp., 𝐺₂) is 𝑛 (resp., 𝑚) and the points of 𝐺₁×𝐺₂ are ordered lexicographically then 𝐻(𝐺₁×𝐺₂)=𝐻(𝐺₁)+_𝐾𝐻(𝐺₂).

Proof. By Lemmas 8 and 11, 𝐻𝐺₁×𝐺₂=𝑟₁+𝑟₂𝐼_𝑚𝑛−𝐴𝐺₁×𝐺₂=𝑟₁+𝑟₂𝐼_𝑚𝑛−𝐴𝐺₁+_𝐾𝐴𝐺₂=𝑟₁+𝑟₂𝐼_𝑚𝑛−𝐴𝐺₁×_𝐾𝐼_𝑚+𝐼_𝑛×_𝐾𝐴𝐺₂=𝑟₁𝐼_𝑚𝑛−𝐼_𝑛×_𝐾𝐴𝐺₂+𝑟₂𝐼_𝑚𝑛−𝐴𝐺₁×_𝐾𝐼_𝑚=𝐼_𝑛×_𝐾𝑟₁𝐼_𝑚−𝐼_𝑛×_𝐾𝐴𝐺₂+𝑟₂𝐼_𝑛×_𝐾𝐼_𝑚−𝐴𝐺₁×_𝐾𝐼_𝑚=𝐼_𝑛×_𝐾𝑟₁𝐼_𝑚−𝐴𝐺₂+𝑟₂𝐼_𝑛−𝐴𝐺₁×_𝐾𝐼_𝑚=𝐼_𝑛×_𝐾𝐻𝐺₂+𝐻𝐺₁×_𝐾𝐼_𝑚=𝐻𝐺₁+_𝐾𝐻𝐺₂, (17) where 𝐼_𝑚𝑚 is the 𝑚𝑛×𝑚𝑛 identity matrix.

Lemma 13. If 𝐴 and 𝐵 are triangulable matrices then the eigenvalues of 𝐴+_𝑘𝐵 are given by 𝛼+𝛽, respectively, as 𝛼 and 𝛽 vary through the eigenvalues of 𝐴 and 𝐵.

Proof. Since 𝐴 and 𝐵 are triangulable, there exist invertible matrices 𝑄 and 𝑃 such that 𝐴₁=𝑄𝐴𝑄⁻¹ and 𝐵₁=𝑃𝐵𝑃⁻¹ are upper triangular. If 𝐴 and 𝐵 are 𝑛×𝑛 and 𝑚×𝑚 matrices, respectively, by Lemmas 9 and 10, 𝐴₁+_𝑘𝐵₁=𝐴₁×_𝐾𝐼_𝑚+𝐼_𝑛×_𝐾𝐵₁=𝑄𝐴𝑄⁻¹×_𝐾𝑃𝐼_𝑚𝑃⁻¹+𝑄𝐼_𝑛𝑄⁻¹×_𝐾𝑃𝐵𝑃⁻¹=𝑄×_𝐾𝑃𝐴𝑄⁻¹×_𝐾𝐼_𝑚𝑃⁻¹+𝑄×_𝐾𝑃𝐼_𝑛𝑄⁻¹×_𝐾𝐵𝑃⁻¹=𝑄×_𝐾𝑃𝐴×_𝐾𝐼_𝑚𝑄⁻¹×_𝐾𝑃⁻¹+𝑄×_𝐾𝑃𝐼_𝑛×_𝐾𝐵𝑄⁻¹×_𝐾𝑃⁻¹=𝑄×_𝐾𝑃𝐴×_𝐾𝐼_𝑚+𝐼_𝑛×_𝐾𝐵𝑄⁻¹×_𝐾𝑃⁻¹=𝑄×_𝐾𝑃𝐴×_𝐾𝐼_𝑚+𝐼_𝑛×_𝐾𝐵𝑄×_𝐾𝑃⁻¹=𝑄×_𝐾𝑃𝐴+_𝑘𝐵𝑄×_𝐾𝑃⁻¹. (18) So 𝐴+_𝑘𝐵 is similar to 𝐴₁+_𝑘𝐵₁ and they have the same eigenvalues. Obviously 𝐴₁+_𝑘𝐵₁=𝐴₁×_𝐾𝐼_𝑚+𝐼_𝑛×_𝐾𝐵₁ is upper triangular with diagonal entries given by 𝛼+𝛽, respectively, as 𝛼 and 𝛽 vary through the eigenvalues of 𝐴₁ and 𝐵₁. Hence the eigenvalues of 𝐴+_𝑘𝐵 are 𝛼+𝛽, respectively, as 𝛼 and 𝛽 vary through the eigenvalues of 𝐴 and 𝐵.

Theorem 14. Let 𝐺₁ and 𝐺₂ be the regular graphs with degrees 𝑚 and 𝑛, respectively. If the eigenvalues of the adjacency matrix 𝐴(𝐺₁) are written as 𝛼₁≤𝛼₂⋯≤𝛼_𝑛−1≤𝛼_𝑚=𝑚 and the eigenvalues of the adjacency matrix 𝐴(𝐺₂) are written as 𝛽₁≤𝛽₂⋯≤𝛽_𝑛−1≤𝛽_𝑛=𝑛, then the number of spanning trees of the Cartesian product 𝐺₁×𝐺₂ is 𝑡𝐺₁×𝐺₂=1𝑚𝑛_𝑖,𝑗(𝑚+𝑛)−𝛼_𝑖+𝛽_𝑗, (19) where 𝑖 and 𝑗 satisfy (𝑖,𝑗)∈{1,2,…,𝑚}×{1,2,…,𝑛}−{(𝑚,𝑛)}.

Proof. We know 𝐺₁×𝐺₂ has 𝑚𝑛 points and the degree of 𝐺₁×𝐺₂ is 𝑚+𝑛. By Lemma 11, 𝐴(𝐺₁×𝐺₂)=𝐴(𝐺₁)+_𝐾𝐴(𝐺₂). By Lemma 13, the eigenvalues of 𝐴(𝐺₁×𝐺₂) are 𝛼_𝑖+𝛽_𝑗 for 𝑖=1,2,…,𝑚 and 𝑗=1,2,…,𝑛. The result follows by Lemma 5.

Theorem 15. If 𝐺₁ and 𝐺₂ are the regular graph of degrees 𝑚 and 𝑛, respectively, the eigenvalues of the Kirchhoff matrix 𝐻(𝐺₁) are written as 0=𝜆₀≤𝜆₁⋯≤𝜆_𝑚−1, and the eigenvalues of the Kirchhoff matrix 𝐻(𝐺₂) are written as 0=𝛾₀≤𝛾₁⋯≤𝛾_𝑛−1, then the number of spanning trees of the Cartesian product of 𝐺₁ and 𝐺₂ is 𝑡𝐺₁×𝐺₂=1𝑚𝑛_𝑖,𝑗𝜆_𝑖+𝛾_𝑗, (20) where 𝑖 and 𝑗 satisfy (𝑖,𝑗)∈{0,1,…,𝑚−1}×{0,1,…,𝑛−1}−{(0,0)}.

Proof. By Lemma 12, 𝐻(𝐺₁×𝐺₂)=𝐻(𝐺₁)+_𝐾𝐻(𝐺₂). By Lemma 13, the eigenvalues of 𝐻(𝐺₁×𝐺₂) are 𝜆_𝑖+𝛾_𝑗 for 𝑖=0,1,…,𝑚−1 and 𝑗=0,1,…,𝑛−1. Hence by Lemma 4, the result follows.

5. The Number of Spanning Trees of the 𝑟^𝑛-Lattice Network

Definition 16 (see [17, 18]). The 𝑟^𝑛-lattice networks 𝑅(𝑟,𝑛) are defined as 𝑅()=𝐾𝑟,𝑛^𝑟𝑅()for𝑛=1𝑟,𝑛−1×𝐾_𝑟for𝑛≥2, (21) where 𝐾_𝑟 is a complete graph of 𝑟 points.
When 𝑟=2, 𝑅(2,𝑛) is well known, the Boolean 𝑛-cube network.
We denote 𝑅(𝑟,𝑛)=𝐾_𝑟⁽¹⁾×𝐾_𝑟⁽²⁾×⋯×𝐾_𝑟^(𝑛).

Lemma 17. The eigenvalues of 𝐻(𝐾_𝑟) are 𝑟 with multiplicity 𝑟−1 and 0 with multiplicity 1.

Proof. Since 𝐻(𝐾_𝑟)=𝑟𝐼_𝑟−𝐽_𝑟, where 𝐽_𝑟 is the matrix of all ones, letting 𝑃_𝐾𝑟(𝜆) be the character polynomial of 𝐻(𝐾_𝑟), we obtain by Gaussian elimination 𝑃_𝐾𝑟(𝜆)()=𝜆111⋯10𝜆−𝑟−1⋯⋯00𝜆−𝑟⋯⋯⋮⋮⋮⋱⋯0000𝜆−𝑟=𝜆𝜆−𝑟^𝑟−1. (22) Hence the result follows.

Lemma 18. If the distinct eigenvalues of the Kirchhoff matrix 𝐻(𝑅(𝑟,𝑛)) are 0=𝜆₀≤𝜆₁⋯≤𝜆_𝑛−1≤𝜆_𝑛 then𝜆₀()𝜆=𝑤𝑖𝑡ℎ𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦𝐶𝑛,𝑛=1₁()𝐶()𝜆=𝑟𝑤𝑖𝑡ℎ𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦𝑟−1𝑛,𝑛−1₂=2𝑟𝑤𝑖𝑡ℎ𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦(𝑟−1)²𝐶()⋮𝜆𝑛,𝑛_𝑛−1=(𝑛−1)𝑟𝑤𝑖𝑡ℎ𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦(𝑟−1)^𝑛−1𝐶()𝜆𝑛,1_𝑛=𝑛𝑟𝑤𝑖𝑡ℎ𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦(𝑟−1)^𝑛𝐶(𝑛,0), (23)
where 𝐶(𝑛,𝑟)=𝑛!/𝑟!(𝑛−𝑟)!.

Proof. Since 𝑅(𝑟,𝑛)=𝐾_𝑟⁽¹⁾×𝐾_𝑟⁽²⁾×⋯×𝐾_𝑟^(𝑛)and by Lemma 13, we obtain each of eigenvalues of 𝐻(𝑅(𝑟,𝑛))=∑^𝑛_𝑖=1one of eigenvalues of 𝐻(𝐾_𝑟^(𝑖)).
Hence if we take the eigenvalue 0 of 𝐻(𝐾_𝑟^(𝑖)) for 𝑖=1,2,…,𝑛 then 𝜆₀=0 with multiplicity 𝐶(𝑛,𝑛)=1. If we take the eigenvalue 𝑟 of 𝐻(𝐾_𝑟^(𝑙)) for some one 𝑙∈{1,2,…,𝑛} and the eigenvalue 0 of 𝐻(𝐾_𝑟^(𝑖)) for each 𝑖∈{1,2,…,𝑛}−{𝑙} then 𝜆₁=𝑟 with multiplicity (𝑟−1)𝐶(𝑛,𝑛−1). If we take the eigenvalue 𝑟 of 𝐻(𝐾_𝑟^(𝑙)) and 𝐻(𝐾_𝑟^(𝑚)), respectively, for 𝑙,𝑚∈{1,2,…,𝑛} and the eigenvalue 0 of 𝐻(𝐾_𝑟^(𝑖)) for each 𝑖∈{1,2,…,𝑛}−{𝑙,𝑚}, then 𝜆₂=2𝑟 with multiplicity (𝑟−1)²𝐶(𝑛,𝑛). 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 𝑅(𝑟,𝑛) is 𝑡(𝑅(𝑟,𝑛))=𝑟^{𝑟𝑛𝑛−𝑛−1}_𝑖=2𝑖^{𝐶(𝑛,𝑖)(𝑟−1)𝑖}. (24)

Proof. Since the degree of 𝑅(𝑟,𝑛) is 𝑛(𝑟−1), the number of points of 𝑅(𝑟,𝑛) is 𝑟^𝑛. By Lemma 18 and Theorem 15, we obtain 𝑡(𝑅(=1𝑟,𝑛))𝑟^𝑛⋅𝑟^{𝐶(𝑛,𝑛−1)(𝑟−1)}⋅(2𝑟)^{2𝐶(𝑛,𝑛−2)(𝑟−1)}⋅(3𝑟)^{3𝐶(𝑛,𝑛−3)(𝑟−1)}⋯(𝑛𝑟)^{𝐶(𝑛,0)(𝑟−1)𝑛}=1𝑟^𝑛⋅𝑟^{𝐶(𝑛,𝑛−1)(𝑟−1)+𝐶(𝑛,𝑛−2)(𝑟−1)2+𝐶(𝑛,𝑛−3)(𝑟−1)3+⋯+𝐶(𝑛,0)(𝑟−1)𝑛}⋅^𝑛_𝑖=1𝑖^{𝐶(𝑛,𝑛−𝑖)(𝑟−1)𝑖}=1𝑟^𝑛⋅𝑟^𝑟𝑛−1⋅^𝑛_𝑖=1𝑖^{𝐶(𝑛,𝑛−𝑖)(𝑟−1)𝑖}=𝑟^{𝑟𝑛𝑛−𝑛−1}_𝑖=2𝑖^{𝐶(𝑛,𝑖)(𝑟−1)𝑖}. (25)

Corollary 20 (see [18]). The number of spanning trees of the Boolean 𝑛-cube network 𝐵_𝑛 is 𝑡𝐵_𝑛=2^{2𝑛𝑛−𝑛−1}_𝑖=2𝑖^{𝐶(𝑛,𝑖)}. (26)

Proof. Since 𝐵_𝑛=𝑅(2,𝑛), by Theorem 15, the result follows.

6. The Number of Spanning Trees of the 2×3⋯×𝑛-Lattice Network

Definition 21. The 2×3⋯×𝑛 lattice network 𝑄_𝑛 can be defined recursively by 𝑄₂=𝐾₂ and 𝑄_𝑛=𝐾_𝑛×𝑄_𝑛−1.
Thus 𝑄_𝑛 has 𝑛! points. We denote 𝑄_𝑛=𝐾₂×𝐾₃×⋯×𝐾_𝑛.

Theorem 22. The number of spanning trees of 𝑄_𝑛 is 𝑡𝑄_𝑛=1𝑛!^𝑛−1_{𝑖=12≤𝑟1<𝑟2<⋯<𝑟𝑖≤𝑛}^𝑖_𝑗=1𝑟_𝑗^{∏𝑖𝑗=1(𝑟𝑗−1)}. (27)

Proof. Since the eigenvalues of 𝐻(𝐾_𝑟) are 𝑟 with multiplicity 𝑟−1 and 0 with multiplicity 1, the distinct ∑^𝑛−1_𝑖=0𝐶(𝑛−1,𝑖)=2^𝑛−1 eigenvalues of 𝑄_𝑛 are 0 and 𝜆_{𝑖,𝑟1,𝑟2,…,𝑟𝑖}=∑^𝑖_𝑗=1𝑟_𝑗 with multiplicity (𝑟₁−1)(𝑟₂−1)⋯(𝑟_𝑖−1), where 𝑟₁,𝑟₂,…,𝑟_𝑖, satisfying 2≤𝑟₁<𝑟₂<⋯<𝑟_𝑖≤𝑛,  𝑖=1,2,…,𝑛−1, are nonzero eigenvalues of 𝐻(𝐾_𝑟1),𝐻(𝐾_𝑟2),…,𝐻(𝐾_𝑟𝑖), respectively, and take zero eigenvalues for the remaining 𝐻(𝐾_𝑟), where 𝑟≠𝑟₁,𝑟₂,…,𝑟_𝑖,  𝑟=2,3,…,𝑛−1. By Lemma 13 and Theorem 15, the result follows.

Example 23. The number of spanning trees of 𝑄₃ and 𝑄₄ is as shown in Figure 1, where 𝑡(𝑄₃)=(1/3!)2⁽²⁻¹⁾3⁽³⁻¹⁾(2+3)^{(2−1)(3−1)}=75 and 𝑡(𝑄₄)=(1/4!)2⁽²⁻¹⁾3⁽³⁻¹⁾4⁽⁴⁻¹⁾(2+3)^{(2−1)(3−1)}(2+4)^{(2−1)(4−1)}(3+4)^{(3−1)(4−1)}(2+3+4)^{(2−1)(3−1)(4−1)} = 1620609272381440.

Figure 1 

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

Definition 24. The generalized Boolean 𝑛-cube network 𝐵𝑅(𝑟,𝑛) can be defined by ()=𝐶𝐵𝑅𝑟,𝑛^𝑟()for𝑛=1𝐵𝑅𝑟,𝑛−1×𝐾₂for𝑛≥2, (28) where 𝐶_𝑟 is a cycle with 𝑟 points. One denotes 𝐵𝑅(𝑟,𝑛)=𝐶_𝑟×𝐾₂×⋯×𝐾₂.
Setting 𝐸_𝑛 is the 𝑛×𝑛 matrix by 𝐸_𝑛=,𝐸0100⋯00010⋯0⋮⋮⋮⋮⋯⋮0000⋯11000⋯0_𝑖𝑛=00⋮00⋯⋯⋮⋯⋯00⋮1000⋮0110⋮0001⋮0000⋮00⋯⋯⋯⋯⋯00⋮00𝑖.1𝑖+1 (29)

Lemma 25. The eigenvalues of the adjacent matrix 𝐴(𝐾_𝑟) are −1 with multiplicity 𝑟−1 and 𝑟−1 with multiplicity 1.

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

Lemma 27. The eigenvalues of the adjacent matrix 𝐴(𝐶_𝑛) are 2cos(2𝜋𝑘/𝑛) for 𝑘=0,1,2,…,𝑛−1.

Proof. Since det𝜆𝐼_𝑛−𝐸_𝑛=𝜆−10⋯00𝜆−100⋮0⋱⋱00⋯0𝜆−1−10⋯0𝜆=𝜆⋅𝜆^𝑛−1+(−1)(−1)^𝑛+1(−1)^𝑛−1=𝜆^𝑛−1, (30) the eigenvalues of 𝐸_𝑛 are 𝑒^{(2𝜋𝑘/𝑛)𝑖} for 𝑘=0,1,2,…,𝑛−1. It follows that 𝐴(𝐶_𝑛)=𝐸_𝑛+𝐸_𝑛^𝑛−1. By Lemma 26, the eigenvalues of 𝐴(𝐶_𝑛) are 𝑒^{(2𝜋𝑘/𝑛)𝑖}+(𝑒^{(2𝜋𝑘/𝑛)𝑖})^𝑛−1=𝑒^{(2𝜋𝑘/𝑛)𝑖}+(𝑒^{2𝜋𝑘𝑖−(2𝜋𝑘/𝑛)𝑖})=𝑒^{(2𝜋𝑘/𝑛)𝑖}+𝑒^{−(2𝜋𝑘/𝑛)𝑖}=2cos(2𝜋𝑘/𝑛) for 𝑘=0,1,2,…,𝑛−1.

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

Theorem 28 (see [13]). The number of spanning trees of 𝐵𝑅(𝑟,𝑛) is 𝑡(𝐵𝑅(𝑟,𝑛))=𝑟2^{𝑟(2𝑛−1−1)−𝑛+1}×^𝑛−1_𝑖=1𝑖^𝑟−1_𝑘=1𝑖+1−cos2𝜋𝑘𝑟^{𝐶(𝑛−1,𝑖)}. (31)

Proof. It follows that the points of 𝐵𝑅(𝑟,𝑛) are 𝑟⋅2^𝑛−1 and the degree of any edge of 𝐵𝑅(𝑟,𝑛) is 𝑛+1. By Lemma 25, the eigenvalues of the adjacent matrix 𝐴(𝐾₂) are −1 and 1. By Lemmas 13 and 27, the distinct eigenvalues of the adjacent matrix 𝐴(𝐵𝑅(𝑟,𝑛)) are (𝑛−2𝑖−1)+2cos2𝜋𝑘𝑟(),𝑘=0,1,…,𝑟−1withmultiplicity𝐶𝑛−1,𝑖 (32) where 𝑖=0,1,…,𝑛−1.
When 𝑘=0 and 𝑖=0, the eigenvalue is 𝑛+1. When 𝑘=0 and 𝑖=0,1,…,𝑛−1, the eigenvalues are (𝑛−2𝑖−1)+2. By Theorem 14, 𝑡((=1𝐵𝑅𝑟,𝑛))𝑟⋅2^{𝑛−1𝑛−1}_𝑖=1((𝑛+1)−((𝑛−2𝑖−1)+2))^{𝐶(𝑛−1,𝑖)}⋅^𝑟−1_𝑘=1^𝑛−1_𝑖=0(𝑛+1)−(𝑛−2𝑖−1)+2cos2𝜋𝑘𝑟^{𝐶(𝑛−1,𝑖)}=1𝑟⋅2^{𝑛−1𝑛−1}_𝑖=1(2𝑖)^{𝐶(𝑛−1,𝑖)}⋅^𝑟−1_𝑘=1^𝑛−1_𝑖=02𝑖+1−cos2𝜋𝑘𝑟^{𝐶(𝑛−1,𝑖)}. (33) Since ∏^𝑛−1_𝑖=02^{𝐶(𝑛−1,𝑖)}=2^{∑𝑛−1𝑖=0𝐶(𝑛−1,𝑖)} = 2^2𝑛−1 and ∏^𝑛−1_𝑖=12^{𝐶(𝑛−1,𝑖)}=2^{2𝑛−1−1}, hence 𝑡((=1𝐵𝑅𝑟,𝑛))𝑟⋅2^𝑛−12^{2𝑛−1−1}2^{(𝑟−1)2𝑛−1𝑛−1}_𝑖=1𝑖^{𝐶(𝑛−1,𝑖)}⋅^𝑟−1_𝑘=1^𝑛−1_𝑖=0𝑖+1−cos2𝜋𝑘𝑟^{𝐶(𝑛−1,𝑖)}. (34) Since ∏^𝑟−1_𝑘=1sin²(𝜋𝑘/𝑟)=𝑟² and ∏^𝑟−1_𝑘=1(1−cos(2𝜋𝑘/𝑟))=∏^𝑟−1_𝑘=12sin²(𝜋𝑘/𝑟)=𝑟²/2^𝑟−1 as 𝑖=1, 𝑡(𝐵𝑅(𝑟,𝑛))=𝑟2^{𝑟(2𝑛−1−1)−𝑛+1𝑛−1}_𝑖=1𝑖^{𝐶(𝑛−1,𝑖)}⋅^𝑟−1_𝑘=1^𝑛−1_𝑖=1𝑖+1−cos2𝜋𝑘𝑟^{𝐶(𝑛−1,𝑖)}=𝑟2^{𝑟(2𝑛−1−1)−𝑛+1}×^𝑛−1_𝑖=1𝑖^𝑟−1_𝑘=1𝑖+1−cos2𝜋𝑘𝑟^{𝐶(𝑛−1,𝑖)}. (35)

8. The Number of Spanning Trees of the Hypercube Network

Definition 29. The hypercube network 𝐻(𝑟,𝑚) can be defined by 𝐻()=𝐶𝑟,𝑚^𝑟𝐻()for𝑚=1𝑟,𝑚−1×𝐶_𝑟for𝑚≥2, (36) where 𝐶_𝑟 is a cycle with 𝑟 points.

Theorem 30. The number of spanning trees of 𝐻(𝑟,𝑚) is 𝑡(𝐻(=2𝑟,𝑚))^𝑟𝑚−1𝑟^𝑚_{𝑙1,𝑙2,…,𝑙𝑚}𝑚−^𝑚_𝑖=1cos2𝜋𝑙_𝑖𝑟, (37) where 𝑙₁,𝑙₂,…,𝑙_𝑚 satisfy (𝑙₁,𝑙₂,…,𝑙_𝑚)∈{0,1,…,𝑟−1}×{0,1,…,𝑟−1}×⋯×{0,1,…,𝑟−1}−{(0,0,…,0)}.

Proof. It follows that the points of 𝐻(𝑟,𝑚) are 𝑟^𝑚 and the degree of any edge of 𝐻(𝑟,𝑚) is 2𝑚. By Lemma 26 and Theorem 14, 𝑡(𝐻(=1𝑟,𝑚))𝑟^𝑚_{𝑙1,𝑙2,…,𝑙𝑚}2𝑚−2^𝑚_𝑖=1cos2𝜋𝑙_𝑖𝑟=2^𝑟𝑚−1𝑟^𝑚_{𝑙1,𝑙2,…,𝑙𝑚}𝑚−^𝑚_𝑖=1cos2𝜋𝑙_𝑖𝑟, (38) where 𝑙₁,𝑙₂,…,𝑙_𝑚 satisfy (𝑙₁,𝑙₂,…,𝑙_𝑚)∈{0,1,…,𝑟−1}×{0,1,…,𝑟−1}×⋯×{0,1,…,𝑟−1}−{(0,0,…,0)}.

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, 𝐺₁ and 𝐺₂, are given by 𝜆+𝛾 as 𝜆 and 𝛾 vary through the eigenvalues of the Kirchhoff matrices 𝐻(𝐺₁) and 𝐻(𝐺₂), 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.