Abstract
The problem of enumerating spanning trees in lattices with Klein bottle boundary condition is considered here. The exact closed-form expressions of the numbers of spanning trees for 4.8.8 lattice, hexagonal lattice, and 33·42 lattice on the Klein bottle are presented.
1. Introduction
Let denote a graph with no multiple edges and no loops and with vertex set and edge set . The degree of a vertex is the number of edges attached to it. A -regular graph is a graph with the property that each of its vertices has the same degree . The adjacency matrix of is the matrix with elements if and are connected by an edge and zero otherwise. The Laplacian matrix is the matrix with the element , where is the Kronecker delta, equal to 1 if , and zero otherwise. Denote by the number of spanning trees of a graph . Enumeration of spanning trees on the graph is a problem of fundamental interest in mathematics and physics. This number can be calculated in several ways. A basic result is “the Matrix-Tree Theorem.”
Theorem 1 (see [1]). Let be a graph with vertex set and let be its Laplacian matrix. Then, where is the submatrix of by deleting the sth row and the sth column from for .
Note that one of the eigenvalues of is always zero. We can express that can be expressed by the nonzero eigenvalue of as follows.
Lemma 2 (see [1]). Let be the Laplacian eigenvalues of a connected graph with vertices. Then, .
By two methods, Ciucu et al. [2] obtained a factorization theorem for the number of spanning trees of the plane graphs with reflective symmetry (all orbits have two vertices). In [3], Zhang and Yan obtained a factorization theorem for the number of spanning trees of the more general graphs with reflective symmetry (i.e., the so-called graphs with an involution, and all orbits have one or two vertices). A graph is said to be -rotational symmetric if the cyclic group of order is a subgroup of the automorphism group of . Yan and Zhang [4] also obtained a factorization theorem for -rotational symmetric graph. As applications, they got explicit expressions for the numbers of spanning trees and the asymptotic tree number entropy for some lattices with cylindrical boundary condition.
Lattices are of special interest for their structures. In particular, the number of spanning trees in a lattice was studied extensively. It turns out that has asymptotically exponential growth; one defines the quantity by This limit is known as the asymptotic tree number entropy, asymptotic growth constant, or thermodynamical limit.
Closed-form expressions for have been obtained for many lattices. Wu [5] evaluated the number of spanning trees on a large planar lattice, exactly for the square, triangular, and honeycomb lattice. Tzeng and Wu [6] obtained the spanning tree generating function for a hypercubic lattice in dimensions under free, periodic, and a combination of free and periodic boundary conditions and a quartic lattice embedded on a Möbius strip and the Klein bottle. Shrock and Wu [7] got a general formulation for the number of spanning trees on lattices in high dimensions. With the formulation, closed-form expressions for the number of spanning trees for hypercubic, body-centred cubic, face-centred cubic, and specific planar lattices including the kagomé, diced, 4.8.8 (bathroom-tile), Union Jack, and 3.12.12 lattices are obtained. With the same method, Chang and Shrock [8] got closed-form expressions of the number of spanning trees for the -dimensional body-centred cubic lattice and thermodynamical limit. They also gave an exact integral expression for thermodynamical limit on the face-centred cubic lattice and 4.8.8 lattice. Chang and Wang [9] considered the number of spanning trees of some Archimedean lattices and hypercubic lattices. More related results can be found in [10, 11].
In this paper, we present an exact closed-form result for the asymptotic growth constant for spanning trees of lattices embedded on Klein bottle, exactly for 4.8.8 lattice, hexagonal lattice, and lattice. The number of spanning trees of 4.8.8 lattice is gotten in Section 2. With the same method, we consider hexagonal lattice and lattice in Sections 3 and 4, respectively.
2. The 4.8.8 Lattice
Introduce some notation firstly. Let and be the inverse and the transpose of a matrix . And let denote the identity matrix. Set Let be an matrices with entries It is not difficult to check that the elements of the matrices are The entries of the matrices , and are where for .
The 4.8.8 lattice is shown in Figure 1(a). If we add edges , for in , we obtain a graph with cylindrical boundary condition, denoted by . Adding edges , for in , a 4.8.8 lattice with toroidal boundary condition, denoted by , can be gotten.
(a)
(b)
(c)
Yan and Zhang [4] got the number of spanning trees and the asymptotic tree number entropy of : where , , and .
Shrock and Wu [7] showed that the number of spanning trees and the asymptotic tree number entropy of can be expressed as where and . Chang and Shrock [8] obtained a closed-form expression of by an exact closed-form evaluation of the integral given in [7].
By adding edges , for in , 4.8.8 lattice with Klein bottle boundary condition can be gotten. By a suitable labelling of vertices of , the adjacency matrix of it can be written in terms of a linear combination of direct products of smaller ones: where
By (6), we havewhere for ,
is the degree of the vertices of .
Interchanging rows and columns, those matrices can be changed into a block-diagonal form having the same determinants: where .
For an even value of (the case when is odd is similar), the Laplacian characteristic polynomial of can be expressed as where for , and for . Note that Hence, by Lemma 2, where are the nonzero Laplacian eigenvalues of .
Note that the matrix also is a Laplacian matrix of a graph, denoted by (see Figure 2(a)). Then, and . So, we have Formula (18) is also suitable for other lattices on the Klein bottle with similar proof, but making use of different Laplacian matrix. It will be used later on for two other types of lattices, replacing by (see Figure 2(b)) and (see Figure 2(c)), respectively.
(a)
(b)
(c)
In the following, we turn to calculate . Let be a subset of the row/column index set of . For convenience, let denote the determinant of the matrix obtained from by deleting all rows and columns whose indices are in . For , noticing that and , expanding the determinant , along the first row, and then expanding the resulting determinants along the first column, we have
Now, we turn to calculate , , , and .
Let , , , and . Also set , , .
By the Laplace expansion theorem, we obtain several expansions. First, an expansion by rows , and : An expansion by rows , and , we get
The recursion relations (20) and (21) give Note that Making use of the initial conditions, respectively, and solving (22), we obtain where and .
By combining (19) and (24), we obtain Similarly, by calculation, we have
Expanding the determinant along the first row and then expanding the resulting determinants along the first column, we have Thus, we have the following.
Theorem 3. The number of spanning trees of lattice can be expressed as where and .
3. The Hexagonal Lattice
The hexagonal lattice is shown in Figure 1(b). If we identify and and , and and for or in , we obtain a graph with cylindrical boundary condition, denoted by . Adding edges for , in , a hexagonal lattice with toroidal boundary condition, denoted by , can be gotten.
Yan and Zhang [4] got the number of spanning trees and the asymptotic tree number entropy of where , , and .
Shrock and Wu [7] showed that the number of spanning trees and the asymptotic tree number entropy of can be expressed as where and .
By adding edges for , in , a hexagonal lattice with Klein bottle boundary condition can be gotten. For the number of spanning trees of , we have the following result.
Theorem 4. The number of spanning trees of hexagonal lattice can be expressed as where and .
Proof. By suitable labelling of vertices of , the adjacency matrix of it can be written in terms of a linear combination of direct products of smaller ones:
where
, in which if is odd and ; else, , in which , if is even, and ; else, , in which , otherwise .
Interchanging rows and columns, we have
where in which , if is even, and , and ; else, , in which , , otherwise 0. Expanding the determinant along the first row and then expanding the resulting determinants along the first column, we have
Let , , , and . Also, set , , .
By the Laplace expansion theorem, we obtain several expansions. First, an expansion by rows and is as follows:
An expansion by rows and is as follows:
The recursion relations (36) and (37) give
Note that
Making use of the initial conditions, respectively, and solving (38), we obtain
where , and . By combining (35) and (40) we obtain
Similarly, by calculation, we have
Expanding the determinant along the first row and then expanding the resulting determinants along the first column, we have
By formula (18), the result holds.
4. The Lattice
The lattice can be constructed by starting with the square lattice and adding a diagonal edge connecting the vertices in the upper left to the lower right corners of each square in every other row as shown in Figure 1(c). If we identify and , and , and and for , in , we obtain a graph with cylindrical boundary condition, denoted by . Adding edges for , in , a lattice with toroidal boundary condition, denoted by , can be gotten.
Yan and Zhang [4] got the number of spanning trees and the asymptotic tree number entropy of : where , , and .
Chang and Wang [9] showed that the number of spanning trees and the asymptotic tree number entropy of can be expressed as where and .
By adding edges for , in , a lattice with Klein bottle boundary condition can be gotten. For the number of spanning trees of , we have the following theorem.
Theorem 5. The number of spanning trees of lattice can be expressed as where and .
Proof. By a suitable lebelling of vertices of , the adjacency matrix of it can be written in terms of a linear combination of direct products of smaller ones:
where and , where , if is odd, and ; else, , , in which , otherwise . Using the same notations as Section 2, we have
Let , , , and . Also set , , .
By the Laplace expansion theorem, we obtain several expansions. First, an expansion by rows and is as follows:
An expansion by rows 1 and 2 is as follows:
The recursion relations (49) and (50) give
Note that
Making use of the initial conditions, respectively, and solving (51), we obtain
where , and . By combining (48) and (53) we obtain
Similarly, by calculation, we have
When is even, expanding the determinant along the first row and then expanding the resulting determinants along the first column, we have
By formula (18), the result holds.
5. Concluding Remarks
In this paper, we computed the numbers of spanning trees for 4.8.8 lattice, hexagonal lattice, and lattice with a Klein bottle boundary condition. For the asymptotic tree number entropy of graphs, Lyons [11] got the following result.
Theorem 6. Let be a tight sequence of finite connected graphs with bounded average degree such that . If is a sequence of connected subgraph of , such that , then .
By Theorem 6 (or compared with the results by Chang and Shrock [8], Chang and Wang [9], Shrock and Wu [7], and Yan and Zhang [4]), we can see that 4.8.8 lattices have the same asymptotic tree number entropy with three different boundary conditions (cylindrical, toroidal, and Klein bottle). Also hexagonal lattice and lattice have the same property.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by Promotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (Grant no. BS2013DX026) and AMEP of Linyi University.