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

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1 Applied Mathematics Volume 20, Article ID , 14 pages doi:101155/20/ 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, Konya, Turkey Correspondence should be addressed to Fatih Yılmaz, Received 9 February 20; Accepted 26 March 20 Academic Editor: Ferenc Hartung Copyright q 20 F Yılmaz and D 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 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 A m 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 kth term of the linear homogeneous recurrence relation with constant coefficients is an equation of the form a nk c 0 a n c 1 a n1 c k 1 a nk 1, 11 where c 0,c 1,,c k 1 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: J n2 J n1 2J n, where J 0 0, J 1 1

2 2 Applied Mathematics for n 0 The first few values of the sequence are n J n Consider a graph G V, E, with set of vertices V G {1, 2,,n} and set of edges EG {e 1,e 2,,e m } 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, a ij { 1, if there is a directed edge connecting ai to a j, 0, otherwise 14 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 11 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 n-square matrix is defined by per A σ S n n a iσi, i1 15 where the summation extends over all permutations σ of the symmetric group S n 6 Let A a ij be an m n matrix with row vectors r 1,r 2,,r m 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 a ik / 0, a jk / 0, and i / j Then the m 1 n 1 matrix A ij:k obtained from A replacing row i with a jk r i a ik r j and deleting row j and column k is called the contraction of A on column k relative to rows i and j IfA is contractible on row k with a ki / 0, a kj / 0, and i / j, then the matrix A k:ij A T ij:k 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 A can be contracted to a matrix B if either B A or if there exist matrices A 0,A 1,,A t t 1 such that A 0 A, A t B and A r is a contraction of A r 1 for r 1, 2,,t 1 One can see that, if A is a nonnegative integer matrix of order n>1andb is a contraction of A 7, then per A per B 16 In 7, the authors consider relationships between the sums of the Fibonacci and Lucas numbers and 1-factors of bipartite graphs

3 Applied Mathematics 3 v 1 v 2 v 3 v 4 v 4k-3 v 4k 2 v 4k 1 v 4k Figure 1 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 J n1 J n 1 J 2 n 1 n 2 n 1 17 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, the author present a combinatorial proof that the wheel W n has L 2n 2 spanning trees, and L n 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 F p n, and their sums, n i1 F pi, 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, jth entry of A r 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

4 4 Applied Mathematics Theorem 21 Let A a ij be the adjacency matrix of the graph given in Figure 1 with n vertices That is, 2, a m,m1, A 1, a s1,s, a ij 1, a 2p,3p, 0, otherwise, 21 where m 1, 3, 5,,n 1, p 0, 4, 8,,n 4, s {1, 2, 3,,n} {4, 8,,4k}, and k 1, 2,,n/4 Then, a r 1p,1p ar 4p,4p 1 ( 2 2 r 2 r r 4 ), a r 2p,1p ar 4p,3p r 2 r1 2 1 r1 2, a r 3p,1p ar 4p,2p r 2 r1 2 1 r1 2, a r 4p,1p 1 2 r 2 r 2 1 r 2, a r A r 1p,2p ar 3p,4p r 2 r2 4 1 r 4, a r 2p,2p ar 3p,3p r 2 r2 2 1 r1 2, a r 2p,3p ar 3p,2p r 2 r2 2 1 r1 2, a r 1p,3p ar 2p,4p r 2 r2 4 1 r 4, a r 1p,4p r 2 r2 8 1 r 8, 0, otherwise 22 Proof It is known that the rth r N power of a matrix A is computed by using the known expression A r TJ r T 1 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 A λi 0, where I is the identity matrix of nth order

5 Applied Mathematics 5 Let P n x be the characteristic polynomial of the matrix A which is defined in 21 Then we can write P 4 x x 4 5x 2 4 P 8 x x 8 10x 6 33x 4 40x 2 16 P x x 15x 10 87x 8 245x 6 348x 4 240x Taking 23 into account, we obtain P n λ ( λ 4 5λ 2 4) k λ 1λ 1λ 2λ 2 k, 24 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 J k diag 2,, 2, }{{} 2,,2, }{{} 1,, 1, }{{} 1,,1 }{{} 25 Let us consider the relation J T 1 AT AT TJ; here A is nth-order matrix 21, 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 T j Then T T 1,T 2,,T n and AT 1,,AT n T 1 λ 1,,T k λ 1,T k1 λ 2,,T 2k λ 2,,T 3k1 λ 4,,T 4k λ 4 26

6 6 Applied Mathematics In other words, AT 1 T 1 λ 1 AT 2 T 2 λ 1 AT k T k λ 1 AT k1 T k1 λ 2 AT k2 T k2 λ 2 AT 2k T 2k λ 2 AT 2k1 T 2k1 λ 3 27 AT 2k2 T 2k2 λ 3 AT 3k T 3k λ 3 AT 3k1 T 3k1 λ 3 AT 3k2 T 3k2 λ 3 AT 4k T 4k λ 4 Solving the set of equations system, we obtain eigenvectors of the matrix A: T } {{ } } {{ } } {{ } } {{ } n n 28

7 Applied Mathematics 7 We will find inverse matrix T 1 denoting the ith row of the inverse matrix T 1 by T 1 t 1,t 2,,t n and implementing the necessary transformations, we obtain T Using the derived equalities and matrix multiplication, A TJT 1 A r TJ r T 1 a r i,j 210 We obtain the expression for the rth power of the matrix A as in 22, thatis, a r i1,i J r A r a r i,i1 2J r a r 1p,4p 4J r 1 a r 2p,3p ar 3p,2p J r1 if r is odd a r 4p,1p J r 1 0, otherwise a r 14k 1,14k 1 ar 4k,4k 2J r a r 2p,2p ar 3p,3p J r1 A r a r 1p,3p ar 2p,4p 2J r if r is even, a r 3p,1p ar 4p,2p J r 0, otherwise where i 1, 3, 5,,n 1, p 0, 4, 8,,4k 1, and k 1, 2,,n/4

8 8 Applied Mathematics See Appendix B Let us consider the matrix A for n 4asbelow: A One can see that A l 0 2J l 0 4J l 1 J l 0 J l1 0 0 J l1 0 2J l, J l 1 0 J l 0 At 2J t 1 0 2J t 0 0 J t1 0 2J t J t 0 J t1 0 0 J t 0 2J t 1, 213 where l is positive odd integer and t is a positive even integer Then we will give the following corollary without proof Corollary 22 Let A be a matrix as in 2 Then, det A r det A r ( J 2 r J r 1 J r1 ) 2 4 r We call this property as Cassini-like formula for Jacobsthal numbers This formula also is equal to square of the formula given by 17 3 Determinantal Representations of the Jacobsthal Numbers Let H n h 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, H n Theorem 31 Let H n be an n-square matrix n >2 as in 31,then per H n per H n 2 n J n1, 32 where J n is the nth Jacobsthal number

9 Applied Mathematics 9 Proof By definition of the matrix H n, it can be contracted on column 1 Let H r n contraction of H n Ifr 1, then be the rth H 1 n Since H 1 n also can be contracted according to the first column, H 2 n Going with this process, we have H 3 n Continuing this method, we obtain the rth contraction H r n J r1 2J r 0,

10 10 Applied Mathematics where 2 r n 4 Hence H n 3 n J n 2 2J n which, by contraction of H n 3 n on column 1, becomes H n 2 n per H n per H n 2 n J n1 See Appendix A Jn 1 2J n 2 By 16, we have 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 is a r 11 ar 44 ar 55 ar 88 1 ( 2 2 r 2 r r 4 ), a r 21 ar 43 ar 65 ar r 2 r1 2 1 r1 2, a r 31 ar 42 ar 75 ar r 2 r1 2 1 r1 2, a r 41 ar r 2 r 2 1 r 2, a r A r ar 34 ar 56 ar r 2 r2 4 1 r 4, a r 22 ar 33 ar 66 ar r 2 r2 2 1 r1 2, a r 23 ar 32 ar 67 ar r 2 r2 2 1 r1 2, a r 13 ar 24 ar 57 ar r 2 r2 4 1 r 4, a r 14 ar r 2 r2 8 1 r 8, 0, otherwise 41

11 Applied Mathematics 11 For r 4, a 4 11 a4 44 a4 55 a a 4 21 a4 43 a4 65 a ( ) , , a 4 31 a4 42 a4 75 a , a 4 41 a , a 4 A 4 a4 34 a4 56 a a 4 22 a4 33 a4 66 a , , 42 a 4 23 a4 32 a4 67 a , otherwise , a 4 13 a4 24 a4 57 a , a 4 14 a , In other words, A

12 Applied Mathematics For r 5, a 5 11 a5 44 a5 55 a a 5 21 a5 43 a5 65 a a 5 31 a5 42 a5 75 a ( ) , , , a 5 41 a , a 5 a5 34 a5 56 a A 5 a 5 22 a5 33 a5 66 a , , 44 a 5 23 a5 32 a5 67 a , otherwise , a 5 13 a5 24 a5 57 a , a 5 14 a , That is, A 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

13 Applied Mathematics 13 Appendices A Procedure for Contraction Method We give a Maple 13 source code to find permanents of one type of contractible tridiagonal matrix: restart: withlinearalgebra: contraction:procn local i, j, k, c, C; c:i, j piecewisei j 1, 1,j i 1, 2,j 2andi 2, 1,j n and i n, 3,i j, 1; C:Matrixn, n, c: for k from 0 to n 3do printk, C: for j from2ton k do C1,j : C2, 1 C1,jC1, 1 C2,j : od: C:DeleteRowDeleteColumnMatrixn k, n k, C, 1, 2: od: printk,evalc: end proc: B Computation of Matrix Power We give a Maple 13 formula to compute integer powers of the matrix given by 21: withlinalg : r : 1: >a1:2 2 r 2 r 14 1 r 4/, a1 : 0 >a2: 2 2 r 2 r 12 1 r 12/, a2 : 1 >a3:2 2 r 2 r 12 1 r 1 2/, a3 : 0 >a4: 2 r 2 r 2 1 r 2/, a4 : 0 >a5: 4 2 r 2 r r 4/, a5 : 2 >a6:4 2 r 2 r r 12/, a6 : 0 >a7: 4 2 r 2 r r 1 2/, a7 : 1 >a8:4 2 r 2 r r 4/, a8 : 0 >a9: 4 2 r 2 r 28 1 r 8/, a9 : 0 >A4: matrix4, 4, a1,a5,a8,a9,a2,a6,a7,a8,a3,a7,a6,a5,a4,a3,a2,a1, A : BlockDiagonalA4,,A4

14 14 Applied Mathematics Acknowledgment This research is supported by Selcuk University Research Project Coordinatorship BAP References 1 D Kalman, Generalized Fibonacci numbers by matrix methods, The Fibonacci Quarterly, vol 20, no 1, pp 73 76, R B Bapat, Graphs and Matrices, Springer, London, UK, 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 , 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 , H Kıyak, I Gürses, F Yılmaz, and D Bozkurt, A formula for computing integer powers for one type of tridiagonal matrix, Hacettepe Mathematics and Statistics, vol 39, no 3, pp , H Minc, Permanents, vol 6 of Encyclopedia of Mathematics and its Applications, Addison-Wesley, 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, F Yilmaz and D Bozkurt, Hessenberg matrices and the Pell and Perrin numbers, Number Theory, vol 131, no 8, pp , F Koken and D Bozkurt, On the Jacobsthal numbers by matrix methods, International Contemporary Mathematical Sciences, vol 3, no 13, pp , G B Djordjević and H M Srivastava, Incomplete generalized Jacobsthal and Jacobsthal-Lucas numbers, Mathematical and Computer Modelling, vol 42, no 9-10, pp , M Startek, A Włoch, and I Włoch, Fibonacci numbers and Lucas numbers in graphs, Discrete Applied Mathematics, vol 157, no 4, pp , 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, E Kilic and A P Stakhov, On the Fibonacci and Lucas p-numbers, their sums, families of bipartite graphs and permanents of certain matrices, Chaos, Solitons and Fractals, vol 40, no 5, pp , J Clark and D A Holton, A First Look at Graph Theory, World Scientific, R A Horn and C R Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985

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