Journal of Applied Mathematics

Journal of Applied Mathematics / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 673270 | 25 pages | https://doi.org/10.1155/2013/673270

Complexity of Products of Some Complete and Complete Bipartite Graphs

Academic Editor: Roberto Barrio
Received02 Jun 2013
Revised29 Jul 2013
Accepted12 Aug 2013
Published27 Nov 2013

Abstract

The number of spanning trees in graphs (networks) is an important invariant; it is also an important measure of reliability of a network. In this paper, we derive simple formulas of the complexity, number of spanning trees, of products of some complete and complete bipartite graphs such as cartesian product, normal product, composition product, tensor product, and symmetric product, using linear algebra and matrix analysis techniques.

1. Introduction

In this work we deal with simple and finite undirected graphs, whereis the vertex set andis the edge set. For a graph, a spanning tree inis a tree which has the same vertex set as. The number of spanning trees in, also called the complexity of the graph, denoted by, is a well-studied quantity (for long time). A classical result of Kirchhoff [1], can be used to determine the number of spanning trees for. Let; then the Kirchhoff matrixdefined as, characteristic matrix,, whereis the diagonal matrix whose elements are the degrees of the vertices of. Whileis the adjacency matrix ofis defined as follows:(i)andare adjacent and,(ii)equals the degree of vertexif,(iii)otherwise. All of the cofactors ofare equal to. There are other methods for calculating. Letdenote the eigenvalues ofmatrix of apoint graph. Then it is easily shown that. Furthermore, Kelmans and Chelnokov [2] have shown that. The formula for the number of spanning trees in a d-regular graphcan be expressed aswhereare the eigenvalues of the corresponding adjacency matrix of the graph. However, for a few special families of graphs there exist simple formulas that make it much easier to calculate and determine the number of corresponding spanning trees especially when these numbers are very large. One of the first results is due to Cayley [3] who showed that the complete graph onvertices,hasspanning trees,. Another result is that, whereis the complete bipartite graph with bipartite sets containingandvertices, respectively. It is well known, as in, for example, [4, 5]. Another result is due to Sedlek [6] who derived a formula for the wheel onvertices,; he showed that, for. Sedlacek [7] also later derived a formula for the number of spanning trees in a Mobius ladder,,for. Another class of graphs by Boesch et al., for which an explicit formula has been derived, is based on a prism [8, 9].

Now, we can introduce the following lemmas.

Lemma 1 (see [10]). Considerwhereandare the adjacency and degree matrices ofand the complement of, respectively, andis theunit matrix.

Lemma 2. Letbematrix,such that Then,

Proof. From the definition of the circulant determinants, we have

We can generalize the previous lemma as follows.

Lemma 3. Letandsuch that Then,

Lemma 4 (see [11]). Let, let, letand letassume thatare nonsingular matrices. Then

Formulas in Lemmas 2, 3, and 4 give some sort of symmetry in some matrices which facilitates our calculation of determinants.

2. Number of Spanning Trees of Cartesian Product of Graphs

The Cartesian product,, is the simple graph with vertex setand edge setsuch that two verticesandare adjacent inif and only if eitherandis adjacent toinoris adjacent toinand[12].

Theorem 5. Forwe have

Proof. Applying Lemma 1, we have
Thus, In particular,

Theorem 6. For, we have

Proof. Applying Lemma 1, we haveThus,
Using Lemma 2, we have In particular,

3. Number of Spanning Trees of Normal Product of Graphs

The normal product, or the strong product,, is the simple graph withwhereandare adjacent inif and only if eitherandis adjacent to,is adjacent toand, oris adjacent toandis adjacent to[13].

Theorem 7. Forwe have

Proof. Applying Lemma 1, we have
In particular,

Theorem 8. For, we have

Proof. Applying Lemma 1, we have
Using Lemma 2, we have In paricular,

4. Number of Spanning Trees of Composition Product of Graphs

The composition, or lexicographic product,, is the simple graph withas the vertex set in which the verticesandare adjacent if eitheris adjacent toorandis adjacent toin[13].

Theorem 9. For, we have

Proof. Applying Lemma 1, we have
In particular,

Theorem 10. For, we have

Proof. Applying Lemma 1, we haveUsing Lemma 2, we have In particular,

5. Complexity of Tensor Product of Graphs

The tensor product, or Kronecker product,, is the simple graph withwhereandare adjacent inif and only if