Abstract
Using the composition of some existing smaller graphs to construct some large graphs, the number of spanning trees and the Laplacian eigenvalues of such large graphs are also closely related to those of the corresponding smaller ones. By using tools from linear algebra and matrix theory, we establish closed formulae for the number of spanning trees of the composition of two graphs with one of them being an arbitrary complete 3-partite graph and the other being an arbitrary graph. Our results extend some of the previous work, which depend on the structural parameters such as the number of vertices and eigenvalues of the small graphs only.
1. Introduction
In reliable network synthesis, given the class of all connected graphs with vertices and edges, it is very important to seek graphs (known as the -optimal graphs) with the most number of spanning trees, so the number of spanning trees is closely connected to reliable network design [1, 2]. When using a probabilistic graph to model a communication network, the reliability of a network can be expressed as a function of the number of connected spanning subgraphs (spanning trees) of different orders. Thus, the number of spanning trees of a graph describing a network is one of the most natural characteristics for its reliability, and deriving closed formulae of the number of spanning trees for various graphs has attracted the attention of a lot of researchers [3–5].
It is well known that the number of spanning trees of some specific family of graphs can be given explicitly, which include the complete graph , the path , the cycle , the wheel , and the Möbius ladders; “almost-complete” graphs, the threshold graphs, and some multicomplete/star related graphs can also be obtained [6–9].
Mathematic structures can be properly understood if one has a grasp of their symmetries; it also helps to know whether they can be constructed from smaller constituents, since many large graphs (networks) are usually composed from some existing smaller graphs (networks) through graph operations (say, product [10]). In this paper, we are mainly concerned with the number of spanning trees of the composition of two graphs. Let and be two simple graphs; the composition graphs of and , denoted by , are a graph with vertex set , and there is an edge between and if there is an edge connecting and in , or if and there is an edge connecting and in . Sometimes it is also called lexicographic product. The topological parameters and some properties of such large graphs (networks) are associated strongly with those of the corresponding smaller ones.
Since the composition (lexicographic product) of two graphs is noncommutative, for example see Figure 1.
The structure of is different from the structure of . However, the number of the spanning trees of the composition graphs is got in [11]. In this paper, we enumerate spanning trees of the composition graphs with one of them being an arbitrary complete 3-partite graph and the other an arbitrary graph . The number of the spanning trees of the composition graphs depends only on the number of vertices and eigenvalues of small graphs only.
A major open problem which still remains is to devise a technique that would derive close formulae for and , where and are arbitrary graphs, or one of them is an arbitrary regular graph. Fiedler [12] gave the eigenvalues of nonnegative symmetric matrices where he obtains the description of the eigenspace of some matrix operations. This actually has been applied for some graph product in [13]. There may be some relations between the techniques used in this note. For more details on the composition graphs and matrix operations the reader is referred to [14–16].
2. Preliminaries
We start with fixing some notations. Throughout the paper, let be a simple graph with vertex set and edge set . Let be the -adjacency matrix of , and let be the diagonal matrix with being the degree of the th vertex of . The Laplacian matrix of is defined to be , and the corresponding characteristic polynomial of is denoted by . Since the matrix is symmetric, all its eigenvalues are real. We assume without loss of generality that they are arranged in the nondecreasing order; that is, . The number of spanning trees of is denoted by . For other terminologies and notations which are not defined here, the reader is referred to [17].
Lemma 1 (see [17, 18]). Let be the eigenvalues of the Laplacian matrix of the graph ; then
It is well known that the number of spanning trees of a given graph can be calculated through Kirchhoff’s “Matrix-Tree Theorem.” This is one of the first (and most impressive) contributions of spectral theory.
Lemma 2 (see [19]). Let be graph on vertices. Let denote the eigenvalues of . Then
Let be the all-zero matrix, and let be the all-one matrix, of order . Now we give the main results of this paper.
Theorem 3. Let be a simple graph with vertices. Let , , be the Laplacian eigenvalues of which are arranged in the nondecreasing order. Then, the eigenvalues of are , , , and , where the exponents denote the multiplicities of the eigenvalues, for , , and .
Proof. Let be the Laplacian matrix of the graph . Let be the labels of the graph , where if and only if and , or and , where (assume that ), for . The node of the graph is the lexicographic order among vertex sequences; then we can represent the Laplacian matrix in the following block matrix form:
where the element of the block denotes the adjacency relation of the two nodes and . By the definition of the composition graphs, if and only if ; or and . Thus, we have
That is,
Similarly, since the sum of the elements of every row (resp., column) of Laplacian matrix is zero, we have
Let , where is an matrix, . Then we get
where is an by block matrix with each block being equal to , , , and
So the characteristic polynomial of the Laplacian matrix is
where
and , for .
In (10), according to the property of the Laplacian matrix, add all the columns to the first column; then every element of the first column is equal to . Extract ; then every element of the first column is equal to 1. Using the notation from Lemma 1, we have
where
In order to compute the determinant in (13), we start by subtracting the first column from the th column to the last column, getting
where
for .
It follows from (15) that
where
Let , . According to the property of the determinant, we have
where
for .
Adding all the columns to the first column and extracting the term , let
then
where , for .
Adding the first column to all the other columns, we get
for .
From Lemma 1 and the property of the determinant, notice that , ; substituting these into equality (24), we get the value of the determinant :
for .
Combining (17), (19), and (25), we obtain
Then we get the eigenvalues of from above equality immediately. This completes the proof.
Theorem 4. Let be a graph with Laplacian eigenvalues . Then the number of spanning trees of the composition graphs of the complete -partite graph and the graph is equal to where is the order of .
Proof. It follows directly from Lemma 2 and Theorem 3.
3. Some Corollaries
As direct consequences, we give some corollaries of the above Theorem 4.
Corollary 1 (see [19]). The number of spanning trees of the complete -partite graph is equal to where is the order of .
Proof. According to Theorem 4, let , and note that , , and . The corollary follows immediately from Theorem 4.
Corollary 2 (see [18]). One has the following formulae:
Proof. Note that , . Then the corollary follows immediately from Theorem 4.
Finally, as direct consequences of Theorem 4, we give closed formulae for the number of spanning trees of some new family of graphs.
Corollary 3. Let be a complete graph, a cycle with vertices, and the nonzero eigenvalues of ; then the following results can easily be obtained:
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author is grateful to the two anonymous referees for many friendly and helpful suggestions for improving the presentation of this paper. Project is supported by the National Natural Science Foundation of China (no. 70531030 and no. 10641003), National 973 (no. 2007CB311002 and no. 2013CB329404).