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

The number of spanning trees in graphs or in networks is an important issue. The evaluation of this number not only is interesting from amathematical (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.


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][2][3][4][5][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][8][9][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 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  1 and  which is defined recursively by   =  −1 ×  for  ≥ 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.

The Number of Spanning Trees
Definition 1.Let  be a graph with  points labeled 1, 2, . . ., .The adjacency matrix of , (), is an  ×  matrix with the th row and th column entry given by [ ()]  = { 1 if points  and  are adjacent 0 ortherwise.
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, () = (−1) + det(  ()).

Lemma 3.
If  is an  ×  triangulable matrix, which has  eigenvalues, then the sum of product of any  − 1 eigenvalues of  is the sum of all principal minors of .
On the other hand, where  = [  ] and  () =  for  = 1, 2, . . ., .So we only need to prove that the coefficient of  in det( − ) is the sum of all principal minors of .Let   denote the principal minor of  obtained by removing the th row and th column from .
Since a real symmetric matrix is with the property that the sum of its rows (and its columns) is zero, the rank of () ≤ −1.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 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  = 1, 2, . . .,  − 1, there exists eigenvector We obtain ()  = (−  )  .Thus −  is the eigenvalue of () for  = 1, 2, . . .,  − 1.
Hence the lemma is proved by Lemma 4.

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 {, V} denote edge joining points  and V. Let   = (  ,   ) for  = 1, 2; the Cartesian product of  1 and  2 is defined by 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  2 blocks which are denoted by   for  = 1, 2, . . .,  and  = 1, 2, . . ., .That is, where   is the  ×  matrix for  = Lemma 10.If  and  are invertible then (×  ) Proof.Consider where  is  ×  and  is  × .

The Number of Spanning Trees of the Cartesian Product of Regular Graphs
where [( 1 )]  is the (, ) entry of the adjacent matrix ( 1 ) of  1 and ( 2 ) is the  ×  adjacent matrix of  2 .We know ) is an  ×  matrix; it can be described as an  ×  ( × ) block matrix where 0 × is the  ×  zero matrix.Since ( 1 )×    is an  ×  matrix, it can be described as an  ×  ( × ) block matrix Clearly Lemma 12. Let   be the regular graph of degree   for  = 1, 2; then, the degree of  1 ×  2 is  1 +  2 .If the number of the points of  1 (resp.,  2 ) is  (resp., ) and the points of  1 ×  2 are ordered lexicographically then ( 1 ×  2 ) = ( 1 )+  ( 2 ).
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 .

The Number of Spanning Trees of the 𝑟 𝑛 -Lattice Network
Definition 16 (see [17,18]).The   -lattice networks (, ) are defined as where   is a complete graph of  points.When  = 2, (2, ) is well known, the Boolean -cube network.
Lemma 17.The eigenvalues of (  ) are  with multiplicity  − 1 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
. . .
The main theorem in [17,18] can be obtained much more simply by Theorem 19 as follows.

The Number of Spanning
Proof.Since the eigenvalues of (  ) are  with multiplicity  − 1 and 0 with multiplicity 1, the distinct Example 23.The number of spanning trees of  3 and  4 is as shown in Figure 1, where (

The Number of Spanning Trees of the Generalized Boolean 𝑛-Cube Network
Definition 24.The generalized Boolean -cube network (, ) can be defined by where   is a cycle with  points.One denotes (, ) =   ×  2 × ⋅ ⋅ ⋅ ×  2 .
Setting   is the  ×  matrix by Lemma 26 (see [33]).If  is a sequence matrix,  is an eigenvalue of , and  is a polynomial then () is the eigenvalue of ().
The main theorem in [13] can be obtained much more simply as follows.