Journal of Applied Mathematics

Volume 2012 (2012), Article ID 423163, 14 pages

doi:10.1155/2012/423163

Research Article

The Adjacency Matrix of One Type of Directed Graph and the Jacobsthal Numbers and Their Determinantal Representation

Fatih Yılmaz and Durmuş Bozkurt

Department of Mathematics, Science Faculty, Selcuk University, 42250 Konya, Turkey

Received 9 February 2012; Accepted 26 March 2012

Academic Editor: Ferenc Hartung

Copyright © 2012 Fatih Yılmaz and Durmuş Bozkurt. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we consider one type of directed graph. Then we obtain a general form of the adjacency matrices of the graph. By using the well-known property which states the entry of ( is adjacency matrix) is equal to the number of walks of length from vertex to vertex , we show that elements of th positive integer power of the adjacency matrix correspond to well-known Jacobsthal numbers. As a consequence, we give a Cassini-like formula for Jacobsthal numbers. We also give a matrix whose permanents are Jacobsthal numbers.

1. Introduction

The ()th term of the linear homogeneous recurrence relation with constant coefficients is an equation of the form

where () are constants [1]. Some well-known number sequences are in fact a special form of this difference equation. In this paper, we consider the Jacobsthal sequence which is defined by the following recurrence relation:

for . The first few values of the sequence are

Consider a graph , with set of vertices and set of edges . A digraph is a graph whose edges are directed. In the case of a digraph, you can think of the connections as one-way streets along which traffic can flow only in the direction indicated by the arrow. The adjacency matrix for a digraph has a definition similar to the definition of an adjacency matrix for a graph [2]. In other words,

As it is known, graphs are visual objects. Analysis of large graphs often requires computer assistance. So it is necessary to express graphs via matrices. The difference equations of the form (1.1) can be expressed in a matrix form.

In the literature, there are many special types of matrices which have great importance in many scientific work, for example, matrices of tridiagonal, pentadiagonal, and others. These types of matrices frequently appear in interpolation, numerical analysis, solution of boundary value problems, high-order harmonic spectral filtering theory, and so on. In [3–5], the authors investigate arbitrary integer powers of some type of these matrices.

The permanent of a matrix is similar to the determinant but all of the signs used in the Laplace expansion of minors are positive. The permanent of an -square matrix is defined by

where the summation extends over all permutations of the symmetric group [6].

Let be an matrix with row vectors . We call contractible on column (resp., row) , if column (resp., row) contains exactly two nonzero elements. Suppose that is contractible on column with , and . Then the matrix obtained from replacing row with and deleting row and column is called the contraction of on column relative to rows and . If is contractible on row with , and , then the matrix is called the contraction of on row relative to columns and . Every contraction used in this paper will be according to first column. We know that can be contracted to a matrix if either or if there exist matrices () such that and is a contraction of for . One can see that, if is a nonnegative integer matrix of order and is a contraction of [7], then

In [7], the authors consider relationships between the sums of the Fibonacci and Lucas numbers and 1-factors of bipartite graphs.

In [8], the authors investigate the relationships between Hessenberg matrices and the well-known number sequences Pell and Perrin.

In [9], the authors investigate Jacobsthal numbers and obtain some properties for the Jacobsthal numbers. They also give Cassini-like formulas for Jacobsthal numbers as

In [10], the authors investigate incomplete Jacobsthal and Jacobsthal-Lucas numbers.

In [11], the authors consider the number of independent sets in graphs with two elementary cycles. They described the extremal values of the number of independent sets using Fibonacci and Lucas numbers.

In [1], the authors give a generalization for known sequences and then they give the graph representations of the sequences. They generalize Fibonacci, Lucas, Pell, and Tribonacci numbers and they show that the sequences are equal to the total number of -independent sets of special graphs.

In [12], the author present a combinatorial proof that the wheel has spanning trees, and is the th Lucas number and that the number of spanning trees of a related graph is a Fibonacci number.

In [13], the authors consider certain generalizations of the well-known Fibonacci and Lucas numbers, the generalized Fibonacci, and Lucas -numbers. Then they give relationships between the generalized Fibonacci -numbers , and their sums, , and the 1-factors of a class of bipartite graphs. Further they determine certain matrices whose permanents generate the Lucas -numbers and their sums.

In this paper, we consider the adjacency matrices of one type of disconnected directed graph family given with Figure 1.

Figure 1

Then we investigate relationships between the adjacency matrices and the Jacobsthal numbers. We also give one type of tridiagonal matrix whose permanents are Jacobsthal numbers. Then we give a Maple 13 procedure to verify the result easily.

2. The Adjacency Matrix of a Graph and the Jacobsthal Numbers

In this section, we investigate relationships between the adjacency matrix of the graph given by the Figure 1 and the Jacobsthal numbers. Then we give a Cassini-like formula for Jacobsthal numbers.

The th entry of is just the number of walks of length from vertex to vertex . In other words, the number of walks of length from vertex to vertex corresponds to Jacobsthal numbers [14]. One can observe that all integer powers of are specified to the famous Jacobsthal numbers with positive signs.

Theorem 2.1. Let be the adjacency matrix of the graph given in Figure 1 with vertices. That is,

where , , , and . Then,

Proof. It is known that the th power of a matrix is computed by using the known expression [15], where is the Jordan form of the matrix and is the transforming matrix. The matrices and are obtained using eigenvalues and eigenvectors of the matrix .

The eigenvalues of are the roots of the characteristic equation defined by , where is the identity matrix of th order.

Let be the characteristic polynomial of the matrix which is defined in (2.1). Then we can write

Taking (2.3) into account, we obtain

where . Using mathematical induction method, it can be seen easily. The eigenvalues of the matrix are multiple according to the order of the matrix. Then Jordan's form of the matrix is

Let us consider the relation (); here is th-order matrix (2.1), is the Jordan form of the matrix and is the transforming matrix. We will find the transforming matrix . Let us denote the th column of by . Then and

In other words,

Solving the set of equations system, we obtain eigenvectors of the matrix :

We will find inverse matrix denoting the th row of the inverse matrix by and implementing the necessary transformations, we obtain

Using the derived equalities and matrix multiplication,

We obtain the expression for the th power of the matrix as in (2.2), that is,

where and .

See Appendix B.

Let us consider the matrix for as below:

One can see that

where is positive odd integer and is a positive even integer. Then we will give the following corollary without proof.

Corollary 2.2. Let be a matrix as in (2.12). Then,

We call this property as Cassini-like formula for Jacobsthal numbers. This formula also is equal to square of the formula given by (1.7).

3. Determinantal Representations of the Jacobsthal Numbers

Let be -square matrix, in which the main diagonal entries are 1s, except the second and last one which are −1 and 3, respectively. The superdiagonal entries are 2s, the subdiagonal entries are 1s and otherwise 0. In other words,

Theorem 3.1. Let be an -square matrix as in (3.1), then

where is the th Jacobsthal number.

Proof. By definition of the matrix , it can be contracted on column 1. Let be the th contraction of . If , then

Since also can be contracted according to the first column,

Going with this process, we have

Continuing this method, we obtain the th contraction

where . Hence

which, by contraction of on column 1, becomes.By (1.6), we have .

See Appendix A.

4. Examples

We can find the arbitrary positive integer powers of the matrix , taking into account derived expressions.

For , the arbitrary positive integer power of is

For ,

In other words,

For ,

That is,

5. Conclusion

The basic idea of the present paper is to draw attention to find out relationships between graph theory, number theory, and linear algebra. In this content, we consider the adjacency matrices of one type of graph. Then we compute arbitrary positive integer powers of the matrix which are specified to the Jacobsthal numbers.

Appendices

A. Procedure for Contraction Method

We give a Maple 13 source code to find permanents of one type of contractible tridiagonal matrix:

restart:

with(LinearAlgebra):

contraction:=proc()

local ;

:=()→piecewise( and and );

:=Matrix():

for from 0 to do

print():

for from 2 to do

od:

:=DeleteRow(DeleteColumn(Matrix):

od:

print(,eval()):

end proc:

B. Computation of Matrix Power

We give a Maple 13 formula to compute integer powers of the matrix given by (2.1):

Acknowledgment

This research is supported by Selcuk University Research Project Coordinatorship (BAP).

References

D. Kalman, “Generalized Fibonacci numbers by matrix methods,” The Fibonacci Quarterly, vol. 20, no. 1, pp. 73–76, 1982.

R. B. Bapat, Graphs and Matrices, Springer, London, UK, 2006.

J. Rimas, “On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of even order. I,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 783–787, 2005.

J. Rimas, “On computing of arbitrary positive integer powers for one type of symmetric pentadiagonal matrices of even order,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 582–591, 2008.

H. Kıyak, I. Gürses, F. Yılmaz, and D. Bozkurt, “A formula for computing integer powers for one type of tridiagonal matrix,” Hacettepe Journal of Mathematics and Statistics, vol. 39, no. 3, pp. 351–363, 2010.

H. Minc, Permanents, vol. 6 of Encyclopedia of Mathematics and its Applications, Addison-Wesley, 1978.

E. Kilic and D. Tasci, “On families of bipartite graphs associated with sums of Fibonacci and Lucas numbers,” Ars Combinatoria, vol. 89, pp. 31–40, 2008.

F. Yilmaz and D. Bozkurt, “Hessenberg matrices and the Pell and Perrin numbers,” Journal of Number Theory, vol. 131, no. 8, pp. 1390–1396, 2011.

F. Koken and D. Bozkurt, “On the Jacobsthal numbers by matrix methods,” International Journal of Contemporary Mathematical Sciences, vol. 3, no. 13, pp. 605–614, 2008.

G. B. Djordjević and H. M. Srivastava, “Incomplete generalized Jacobsthal and Jacobsthal-Lucas numbers,” Mathematical and Computer Modelling, vol. 42, no. 9-10, pp. 1049–1056, 2005.

M. Startek, A. Włoch, and I. Włoch, “Fibonacci numbers and Lucas numbers in graphs,” Discrete Applied Mathematics, vol. 157, no. 4, pp. 864–868, 2009.

A. T. Benjamin and C. R. Yerger, “Combinatorial interpretations of spanning tree identities,” Bulletin of the Institute of Combinatorics and its Applications, vol. 47, pp. 37–42, 2006.

E. Kilic and A. P. Stakhov, “On the Fibonacci and Lucas -numbers, their sums, families of bipartite graphs and permanents of certain matrices,” Chaos, Solitons and Fractals, vol. 40, no. 5, pp. 2210–2221, 2009.

J. Clark and D. A. Holton, A First Look at Graph Theory, World Scientific, 1995.

R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.