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 (𝑖,𝑗) entry of 𝐴𝑚 (𝐴 is adjacency matrix) is equal to the number of walks of length 𝑚 from vertex 𝑖 to vertex 𝑗, we show that elements of 𝑚th 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 (𝑛+𝑘)th term of the linear homogeneous recurrence relation with constant coefficients is an equation of the form 𝑎𝑛+𝑘=𝑐0𝑎𝑛+𝑐1𝑎𝑛+1++𝑐𝑘1𝑎𝑛+𝑘1,(1.1) where (𝑐0,𝑐1,,𝑐𝑘1) 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: 𝐽𝑛+2=𝐽𝑛+1+2𝐽𝑛,where𝐽0=0,𝐽1=1(1.2) for 𝑛0. The first few values of the sequence are𝑛123456789𝐽𝑛113511214385171.(1.3)

Consider a graph 𝐺=(𝑉,𝐸), with set of vertices 𝑉(𝐺)={1,2,,𝑛} and set of edges 𝐸(𝐺)={𝑒1,𝑒2,,𝑒𝑚}. 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,𝑎𝑖𝑗=1,ifthereisadirectededgeconnecting𝑎𝑖to𝑎𝑗,0,otherwise.(1.4)

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 [35], 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 𝑛-square matrix is defined byper𝐴=𝜎𝑆𝑛𝑛𝑖=1𝑎𝑖𝜎(𝑖),(1.5)

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

Let 𝐴=[𝑎𝑖𝑗] be an 𝑚×𝑛 matrix with row vectors 𝑟1,𝑟2,,𝑟𝑚. We call 𝐴 contractible on column (resp., row) 𝑘, if column (resp., row) 𝑘 contains exactly two nonzero elements. Suppose that 𝐴 is contractible on column 𝑘 with 𝑎𝑖𝑘0,𝑎𝑗𝑘0, and 𝑖𝑗. Then the (𝑚1)×(𝑛1) matrix 𝐴𝑖𝑗𝑘 obtained from 𝐴 replacing row 𝑖 with 𝑎𝑗𝑘𝑟𝑖+𝑎𝑖𝑘𝑟𝑗 and deleting row 𝑗 and column 𝑘 is called the contraction of 𝐴 on column 𝑘 relative to rows 𝑖 and 𝑗. If 𝐴 is contractible on row 𝑘 with 𝑎𝑘𝑖0,𝑎𝑘𝑗0, and 𝑖𝑗, then the matrix 𝐴𝑘𝑖𝑗=[𝐴𝑇𝑖𝑗𝑘]𝑇 is called the contraction of 𝐴 on row 𝑘 relative to columns 𝑖 and 𝑗. Every contraction used in this paper will be according to first column. We know that 𝐴can be contracted to a matrix 𝐵 if either 𝐵=𝐴 or if there exist matrices 𝐴0,𝐴1,,𝐴𝑡 (𝑡1) such that 𝐴0=𝐴,𝐴𝑡=𝐵 and 𝐴𝑟 is a contraction of 𝐴𝑟1 for 𝑟=1,2,,𝑡1. One can see that, if 𝐴 is a nonnegative integer matrix of order 𝑛>1 and 𝐵 is a contraction of 𝐴 [7], then per𝐴=per𝐵.(1.6)

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 𝐽𝑛+1𝐽𝑛1𝐽2𝑛=(1)𝑛2𝑛1.(1.7)

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 𝑘-independent sets of special graphs.

In [12], the author present a combinatorial proof that the wheel 𝑊𝑛 has 𝐿2𝑛2 spanning trees, and 𝐿𝑛 is the 𝑛th 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 𝑝-numbers. Then they give relationships between the generalized Fibonacci 𝑝-numbers 𝐹𝑝(𝑛), and their sums, 𝑛𝑖=1𝐹𝑝(𝑖), and the 1-factors of a class of bipartite graphs. Further they determine certain matrices whose permanents generate the Lucas 𝑝-numbers and their sums.

In this paper, we consider the adjacency matrices of one type of disconnected directed graph family given with 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 𝐴 of the graph given by the Figure 1 and the Jacobsthal numbers. Then we give a Cassini-like formula for Jacobsthal numbers.

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

Theorem 2.1. Let 𝐴=[𝑎𝑖𝑗] be the adjacency matrix of the graph given in Figure 1 with 𝑛 vertices. That is, 𝐴=𝑎𝑖𝑗=2,𝑎𝑚,𝑚+1,1,𝑎𝑠+1,𝑠,1,𝑎2+𝑝,3+𝑝,0,otherwise,(2.1) where 𝑚=1,3,5,,𝑛1, 𝑝=0,4,8,,𝑛4, 𝑠={1,2,3,,𝑛}{4,8,,4𝑘}, and 𝑘=1,2,,𝑛/4. Then, 𝐴𝑟=𝑎𝑟1+𝑝,1+𝑝=𝑎𝑟4+𝑝,4+𝑝=1122(2)𝑟+2𝑟+1+22(1)𝑟+4,𝑎𝑟2+𝑝,1+𝑝=𝑎𝑟4+𝑝,3+𝑝=1122(2)𝑟+2𝑟+1+2(1)𝑟+1+2,𝑎𝑟3+𝑝,1+𝑝=𝑎𝑟4+𝑝,2+𝑝=1122(2)𝑟+2𝑟+1+2(1)𝑟+12,𝑎𝑟4+𝑝,1+𝑝=112(2)𝑟+2𝑟+2(1)𝑟2,𝑎𝑟1+𝑝,2+𝑝=𝑎𝑟3+𝑝,4+𝑝=1124(2)𝑟+2𝑟+24(1)𝑟+4,𝑎𝑟2+𝑝,2+𝑝=𝑎𝑟3+𝑝,3+𝑝=1124(2)𝑟+2𝑟+22(1)𝑟+1+2,𝑎𝑟2+𝑝,3+𝑝=𝑎𝑟3+𝑝,2+𝑝=1124(2)𝑟+2𝑟+22(1)𝑟+12,𝑎𝑟1+𝑝,3+𝑝=𝑎𝑟2+𝑝,4+𝑝=1124(2)𝑟+2𝑟+24(1)𝑟4,𝑎𝑟1+𝑝,4+𝑝=1124(2)𝑟+2𝑟+2+8(1)𝑟8,0,otherwise.(2.2)

Proof. It is known that the 𝑟th (𝑟) power of a matrix 𝐴 is computed by using the known expression 𝐴𝑟=𝑇𝐽𝑟𝑇1 [15], where 𝐽 is the Jordan form of the matrix and 𝑇 is the transforming matrix. The matrices 𝐽 and 𝑇 are obtained using eigenvalues and eigenvectors of the matrix 𝐴.
The eigenvalues of 𝐴 are the roots of the characteristic equation defined by |𝐴𝜆𝐼|=0, where 𝐼 is the identity matrix of 𝑛th order.
Let 𝑃𝑛(𝑥) be the characteristic polynomial of the matrix 𝐴 which is defined in (2.1). Then we can write 𝑃4(𝑥)=𝑥45𝑥2+4𝑃8(𝑥)=𝑥810𝑥6+33𝑥440𝑥2+16𝑃12(𝑥)=𝑥1215𝑥10+87𝑥8245𝑥6+348𝑥4240𝑥2+64(2.3)
Taking (2.3) into account, we obtain 𝑃𝑛(𝜆)=𝜆45𝜆2+4𝑘=[(𝜆1)(𝜆+1)(𝜆2)(𝜆+2)]𝑘,(2.4) where 𝑛=4𝑘,𝑘=1,2,. 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 𝐴 is 𝐽=𝐽𝑘=diag2,,2,𝑘times2,,2,𝑘times1,,1,𝑘times1,,1𝑘times.(2.5)
Let us consider the relation 𝐽=𝑇1𝐴𝑇 (𝐴𝑇=𝑇𝐽); here 𝐴 is 𝑛th-order matrix (2.1), 𝐽 is the Jordan form of the matrix 𝐴 and 𝑇 is the transforming matrix. We will find the transforming matrix 𝑇. Let us denote the 𝑗th column of 𝑇 by 𝑇𝑗. Then 𝑇=(𝑇1,𝑇2,,𝑇𝑛) and 𝐴𝑇1,,𝐴𝑇𝑛=𝑇1𝜆1,,𝑇𝑘𝜆1,𝑇𝑘+1𝜆2,,𝑇2𝑘𝜆2,,𝑇3𝑘+1𝜆4,,𝑇4𝑘𝜆4.(2.6) In other words, 𝐴𝑇1=𝑇1𝜆1𝐴𝑇2=𝑇2𝜆1𝐴𝑇𝑘=𝑇𝑘𝜆1𝐴𝑇𝑘+1=𝑇𝑘+1𝜆2𝐴𝑇𝑘+2=𝑇𝑘+2𝜆2𝐴𝑇2𝑘=𝑇2𝑘𝜆2𝐴𝑇2𝑘+1=𝑇2𝑘+1𝜆3𝐴𝑇2𝑘+2=𝑇2𝑘+2𝜆3𝐴𝑇3𝑘=𝑇3𝑘𝜆3𝐴𝑇3𝑘+1=𝑇3𝑘+1𝜆3𝐴𝑇3𝑘+2=𝑇3𝑘+2𝜆3𝐴𝑇4𝑘=𝑇4𝑘𝜆4.(2.7) Solving the set of equations system, we obtain eigenvectors of the matrix 𝐴: 𝑇=200002002200200002001100200002001100100001001100020020020020020020010010020020010010010010010010002200200002002200100001002200100001001100100001𝑛×𝑛.𝑘times𝑘times𝑘times𝑘times(2.8) We will find inverse matrix 𝑇1 denoting the 𝑖th row of the inverse matrix 𝑇1 by 𝑇1=(𝑡1,𝑡2,,𝑡𝑛) and implementing the necessary transformations, we obtain 𝑇1=1121222000000000000000012220000000000000000122200000000000000001222000000000000122200000000122200000000122200000000122200000000000000000000000022240000000022240000000022240000000022240000000000002224000000000000000022240000000000000000222400000000000000002224.(2.9) Using the derived equalities and matrix multiplication, 𝐴=𝑇𝐽𝑇1𝐴𝑟=𝑇𝐽𝑟𝑇1=𝑎𝑟𝑖,𝑗.(2.10) We obtain the expression for the 𝑟th power of the matrix 𝐴 as in (2.2), that is, 𝐴𝑟=𝑎𝑟𝑖+1,𝑖=𝐽𝑟𝑎𝑟𝑖,𝑖+1=2𝐽𝑟𝑎𝑟1+𝑝,4+𝑝=4𝐽𝑟1𝑎𝑟2+𝑝,3+𝑝=𝑎𝑟3+𝑝,2+𝑝=𝐽𝑟+1𝑎𝑟4+𝑝,1+𝑝=𝐽𝑟10,otherwiseif𝑟isodd𝐴𝑟=𝑎𝑟1+4(𝑘1),1+4(𝑘1)=𝑎𝑟4𝑘,4𝑘=2𝐽𝑟1𝑎𝑟2+𝑝,2+𝑝=𝑎𝑟3+𝑝,3+𝑝=𝐽𝑟+1𝑎𝑟1+𝑝,3+𝑝=𝑎𝑟2+𝑝,4+𝑝=2𝐽𝑟𝑎𝑟3+𝑝,1+𝑝=𝑎𝑟4+𝑝,2+𝑝=𝐽𝑟0,otherwiseif𝑟iseven,(2.11) where 𝑖=1,3,5,,𝑛1,𝑝=0,4,8,,4(𝑘1),and 𝑘=1,2,,𝑛/4.

See Appendix B.

Let us consider the matrix 𝐴 for 𝑛=4 as below: 𝐴=0200101001020010.(2.12) One can see that𝐴𝑙=02𝐽𝑙04𝐽𝑙1𝐽𝑙0𝐽𝑙+100𝐽𝑙+102𝐽𝑙𝐽𝑙10𝐽𝑙0,𝐴𝑡=2𝐽𝑡102𝐽𝑡00𝐽𝑡+102𝐽𝑡𝐽𝑡0𝐽𝑡+100𝐽𝑡02𝐽𝑡1,(2.13)

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

Corollary 2.2. Let 𝐴 be a matrix as in (2.12). Then, det𝐴𝑟=(det𝐴)𝑟=𝐽2𝑟𝐽𝑟1𝐽𝑟+12=4𝑟1.(2.14) 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 𝐻𝑛=[𝑖𝑗]𝑛×𝑛 be 𝑛-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,𝐻𝑛=121120112112011213.(3.1)

Theorem 3.1. Let 𝐻𝑛 be an 𝑛-square matrix (𝑛>2) as in (3.1), then per𝐻𝑛=per𝐻(𝑛2)𝑛=𝐽𝑛+1,(3.2) where 𝐽𝑛 is the 𝑛th Jacobsthal number.

Proof. By definition of the matrix 𝐻𝑛, it can be contracted on column 1. Let 𝐻(𝑟)𝑛 be the 𝑟th contraction of 𝐻𝑛. If 𝑟=1, then 𝐻(1)𝑛=120112112112013.(3.3) Since 𝐻(1)𝑛 also can be contracted according to the first column, 𝐻(2)𝑛=320112112112013.(3.4) Going with this process, we have 𝐻(3)𝑛=560112112112013.(3.5) Continuing this method, we obtain the 𝑟th contraction 𝐻(𝑟)𝑛=𝐽𝑟+12𝐽𝑟0112112112013,(3.6) where 2𝑟𝑛4. Hence 𝐻(𝑛3)𝑛=𝐽𝑛22𝐽𝑛30112013(3.7) which, by contraction of 𝐻(𝑛3)𝑛 on column 1, becomes𝐻(𝑛2)𝑛=𝐽𝑛12𝐽𝑛213.By (1.6), we have per𝐻𝑛=per𝐻(𝑛2)𝑛=𝐽𝑛+1.

See Appendix A.

4. Examples

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

For 𝑘=2, the arbitrary positive integer power of 𝐴 is𝐴𝑟=𝑎𝑟11=𝑎𝑟44=𝑎𝑟55=𝑎𝑟88=1122(2)𝑟+2𝑟+1+22(1)𝑟+4,𝑎𝑟21=𝑎𝑟43=𝑎𝑟65=𝑎𝑟87=1122(2)𝑟+2𝑟+1+2(1)𝑟+1+2,𝑎𝑟31=𝑎𝑟42=𝑎𝑟75=𝑎𝑟86=1122(2)𝑟+2𝑟+1+2(1)𝑟+12,𝑎𝑟41=𝑎𝑟85=112(2)𝑟+2𝑟+2(1)𝑟2,𝑎𝑟12=𝑎𝑟34=𝑎𝑟56=𝑎𝑟78=1124(2)𝑟+2𝑟+24(1)𝑟+4,𝑎𝑟22=𝑎𝑟33=𝑎𝑟66=𝑎𝑟77=1124(2)𝑟+2𝑟+22(1)𝑟+1+2,𝑎𝑟23=𝑎𝑟32=𝑎𝑟67=𝑎𝑟76=1124(2)𝑟+2𝑟+22(1)𝑟+12,𝑎𝑟13=𝑎𝑟24=𝑎𝑟57=𝑎𝑟68=1124(2)𝑟+2𝑟+24(1)𝑟4,𝑎𝑟14=𝑎𝑟58=1124(2)𝑟+2𝑟+2+8(1)𝑟8,0,otherwise.(4.1) For 𝑟=4,𝐴4=𝑎411=𝑎444=𝑎455=𝑎488=1122(2)4+25+22(1)4+4=6,𝑎421=𝑎443=𝑎465=𝑎487=1122(2)4+25+2(1)5+2=0,𝑎431=𝑎442=𝑎475=𝑎486=1122(2)4+25+2(1)52=5,𝑎441=𝑎485=112(2)4+24+2(1)42=0,𝑎412=𝑎434=𝑎456=𝑎478=1124(2)4+264(1)4+4=0,𝑎422=𝑎433=𝑎466=𝑎477=1124(2)4+262(1)5+2=11,𝑎423=𝑎432=𝑎467=𝑎476=1124(2)4+262(1)52=0,𝑎413=𝑎424=𝑎457=𝑎468=1124(2)4+264(1)44=10,𝑎414=𝑎458=1124(2)4+26+8(1)48=0,0,otherwise.(4.2) In other words,𝐴4=601000000011010000050110000005060000000060100000001101000005011000000506.(4.3) For 𝑟=5,𝐴5=𝑎511=𝑎544=𝑎555=𝑎588=1122(2)5+26+22(1)5+4=0,𝑎521=𝑎543=𝑎565=𝑎587=1122(2)5+26+2(1)6+2=11,𝑎531=𝑎542=𝑎575=𝑎586=1122(2)5+26+2(1)62=0,𝑎541=𝑎585=112(2)5+25+2(1)52=5,𝑎512=𝑎534=𝑎556=𝑎578=1124(2)5+274(1)5+4=22,𝑎522=𝑎533=𝑎566=𝑎577=1124(2)5+272(1)6+2=0,𝑎523=𝑎532=𝑎567=𝑎576=1124(2)5+272(1)62=21,𝑎513=𝑎524=𝑎557=𝑎568=1124(2)5+274(1)54=0,𝑎514=𝑎558=1124(2)5+27+8(1)58=20,0,otherwise.(4.4) That is,𝐴5=022020000011021000000210220000501100000000002202000001102100000021022000050110.(4.5)

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(𝑛)

local 𝑖,𝑗,𝑘,𝑐,𝐶;

𝑐:=(𝑖,𝑗)→piecewise(𝑖=𝑗+1,1,𝑗=𝑖+1,2,𝑗=2 and 𝑖=2,1,𝑗=𝑛 and 𝑖=𝑛,3,𝑖=𝑗,1);

𝐶:=Matrix(𝑛,𝑛,𝑐):

for 𝑘 from 0 to 𝑛3 do

print(𝑘,𝐶):

for 𝑗 from 2 to 𝑛𝑘 do

𝐶[1,𝑗]=𝐶[2,1]𝐶[1,𝑗]+𝐶[1,1]𝐶[2,𝑗]

od:

𝐶:=DeleteRow(DeleteColumn(Matrix(𝑛𝑘,𝑛𝑘,𝐶),1),2):

od:

print(𝑘,eval(𝐶)):

end proc:

B. Computation of Matrix Power

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

𝑤𝑖𝑡(𝑙𝑖𝑛𝑎𝑙𝑔)𝑟=1>𝑎1=(2(2)̂𝑟+2̂(𝑟+1)+4(1)̂𝑟+4)/12,𝑎1=0>𝑎2=(2(2)̂𝑟+2̂(𝑟+1)+2(1)̂(𝑟+1)+2)/12,𝑎2=1>𝑎3=(2(2)̂𝑟+2̂(𝑟+1)+2(1)̂(𝑟+1)2)/12,𝑎3=0>𝑎4=((2)̂𝑟+2̂𝑟+2(1)̂𝑟2)/12,𝑎4=0>𝑎5=(4(2)̂𝑟+2̂(𝑟+2)4(1)̂𝑟+4)/12,𝑎5=2>𝑎6=(4(2)̂𝑟+2̂(𝑟+2)2(1)̂(𝑟+1)+2)/12,𝑎6=0>𝑎7=(4(2)̂𝑟+2̂(𝑟+2)2(1)̂(𝑟+1)2)/12,𝑎7=1>𝑎8=(4(2)̂𝑟+2̂(𝑟+2)4(1)̂𝑟4)/12,𝑎8=0>𝑎9=(4(2)̂𝑟+2̂(𝑟+2)+8(1)̂𝑟8)/12,𝑎9=0>𝐴4=𝑚𝑎𝑡𝑟𝑖𝑥(4,4,[𝑎1,𝑎5,𝑎8,𝑎9,𝑎2,𝑎6,𝑎7,𝑎8,𝑎3,𝑎7,𝑎6,𝑎5,𝑎4,𝑎3,𝑎2,𝑎1]),𝐴=𝐵𝑙𝑜𝑐𝑘𝐷𝑖𝑎𝑔𝑜𝑛𝑎𝑙(𝐴4,,𝐴4).

Acknowledgment

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