k-Lucas numbers and associated bipartite graphs

Linear Algebra and its Applications 320 (2000) 51–61 www.elsevier.com/locate/laa

k-Lucas numbers and associated bipartite graphs聻 Gwang-Yeon Lee Department of Mathematics, Hanseo University, Seosan, Chung-Nam 356-706, South Korea Received 27 January 2000; accepted 1 June 2000 Submitted by R.A. Brualdi

Abstract (k)

(k)

For a positive integer k > 2, the k-Fibonacci sequence {gn } is defined as: g1

(k)

(k)

(k)

(k)

(k)

(k)

(k)

= ··· =

gk−2 = 0, gk−1 = gk = 1 and for n > k > 2, gn = gn−1 + gn−2 + · · · + gn−k . More(k) (k) (k) (k) over, the k-Lucas sequence {ln } is defined as ln = gn−1 + gn+k−1 for n > 1. In this paper, (k)

(k)

we consider the relationship between gn and ln and 1-factors of a bipartite graph. © 2000 Elsevier Science Inc. All rights reserved. Keywords: k-Fibonacci sequence; k-Lucas sequence; 1-factor; Permanent

1. Introduction The well-known Fibonacci sequence {Fn } is defined as F1 = F2 = 1

and for n > 2,

Fn = Fn−1 + Fn−2 .

We call Fn the nth Fibonacci number. The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . We consider a generalization of the Fibonacci sequence which is called the kFibonacci sequence for positive integer k > 2. The k-Fibonacci sequence {gn(k) } is defined as (k)

(k)

g1 = · · · = gk−2 = 0, 聻

(k)

(k)

gk−1 = gk = 1

This paper was partially supported by KOSEF, 2000. E-mail address: [email protected] (G.-Y. Lee).

0024-3795/00/$ - see front matter  2000 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 - 3 7 9 5 ( 0 0 ) 0 0 2 0 4 - 4

52

G.-Y. Lee / Linear Algebra and its Applications 320 (2000) 51–61

and for n > k > 2, (k)

(k)

(k)

gn(k) = gn−1 + gn−2 + · · · + gn−k . (k)

We call gn the nth k-Fibonacci number. By the definition of the k-Fibonacci sequence, we know that (k)

(k)

(k)

(k)

(k)

gk+1 =gk + gk−1 = 1 + 1 = 2, (k)

(k)

gk+2 =gk+1 + gk + gk−1 = 2 + 1 + 1 = 22 , (k) (k) (k) (k) =gk+2 + gk+1 + gk(k) + gk−1 = 22 + 2 + 1 + 1 = 23 , gk+3 .. . (k) (k) (k) + · · · + gk(k) + gk−1 = 2k−3 + · · · + 2 + 1 + 1 = 2k−2 , g2k−2 =g2k−3 (k) (k) (k) g2k−1 =g2k−2 + · · · + gk(k) + gk−1 = 2k−2 + 2k−3 + · · · + 2 + 1 + 1 = 2k−1 . (k)

Thus, we have that gj = 2j −k for j = k, k + 1, . . . , 2k − 1. For example, if k = 2, (2)

then {gn } is the Fibonacci sequence. If k = 5, then the 5-Fibonacci sequence is 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3535, 6930, . . . Let E be the 1 × (k − 1) matrix all of whose entries are ones and let In be the identity matrix of order n. For any k > 2, the fundamental recurrence relation, n > k, (k)

(k)

(k)

gn(k) = gn−1 + gn−2 + · · · + gn−k can be defined by the vector recurrence relation  (k)   (k)  gn+1 gn  (k)   (k)  gn+2   gn+1       ..  = Qk  ..  ,  .   .  (k) (k) gn+k−1 gn+k where Qk =

 0 1

 Ik−1 . E

(1)

(2)

By applying (1), we have  (k)   (k)  gn+1 g1  (k)   (k)    gn+2  n g2     ..  = Qk  ..  .  .   .  (k) (k) gk gn+k In [4–6], relationships between the k-Fibonacci numbers and their associated matrices can be found.

G.-Y. Lee / Linear Algebra and its Applications 320 (2000) 51–61

53

We now introduce k-Lucas sequences. We let Ln represent the nth Lucas number, that is, for n > 1, Ln = Fn−1 + Fn+1 , where F0 = 0. The Lucas sequence {Ln } is 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, . . . One can find applications and properties of the Lucas number in [3]. Let g0(k) = 0. (k) The k-Lucas sequence {ln } is defined by (k) (k) ln(k) = gn−1 + gn+k−1 .

We call ln(k) the nth k-Lucas number. Then we have lj(k) = 2j −1 , j = 1, 2, . . . , k −

1, and lk(k) = 1 + 2k−1 . If k = 2, then ln(2) = Ln . For example, if k = 5, then the 5-Lucas sequence is 1, 2, 4, 8, 17, 32, 63, 124, 244, 480, 943, 1854, 3645, 7166, . . . The permanent of an n-square matrix A = [aij ] is defined by perA =

n X Y

aiσ (i) ,

σ ∈Sn i=1

where the summation extends over all permutations σ in the symmetric group Sn . For any square matrix A and any permutation matrices P and Q, perA = perP AQ. A bipartite graph G is a graph whose vertex set V can be partitioned into two subsets V1 and V2 such that every edge of G joins a vertex in V1 and a vertex in V2 . A 1-factor (or perfect matching) of a graph with 2n vertices is a spanning subgraph of G in which every vertex has degree 1. The enumeration or actual construction of 1-factors of a bipartite graph has many applications, for example, in maximal flow problems and in assignment and scheduling problems. Let A(G) be the adjacency matrix of the bipartite graph G, and let µ(G) denote p the number of 1-factors of G. Then, one can find the following fact in [7]: µ(G) = perA(G). Also, one can find more applications of permanents in [7]. Let G be a bipartite graph whose vertex set V is partitioned into two subsets V1 and V2 such that |V1 | = |V2 | = n. We construct the bipartite adjacent matrix B(G) = [bij ] of G as following: bij = 1 if and only if G contains an edge from vi ∈ V1 to vj ∈ V2 , and 0 otherwise. Then, in [7], the number of 1-factors of bipartite graph G equals the permanent of its bipartite adjacency matrix. Let G and G0 denote two general graphs of order n, and let the adjacency matrices of these graphs be denoted by A and A0 , respectively. Then G and G0 are isomorphic if and only if A is transformable into A0 by simultaneous permutations of the lines of A. Thus, G and G0 are isomorphic if and only if there exists a permutation matrix P of order n such that P T AP = A0 . Let A = [aij ] be an m × n real matrix with row vectors α1 , α2 , . . . , αm . We say A is contractible on column (resp. row) k if column (resp. row) k contains exactly two nonzero entries. Suppose A is contractible on column k with aik = / 0= / aj k and

54

G.-Y. Lee / Linear Algebra and its Applications 320 (2000) 51–61

i= / j. Then the (m − 1) × (n − 1) matrix Aij :k obtained from A by replacing row i with aj k αi + aik αj and deleting row j and column k is called the contraction of A / 0= / akj on column k relative to rows i and j. If A is contractible on row k with aki = and i = / j , then the matrix Ak:ij = [ATij :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 on the first column using the first and second rows. We say that A can be contracted to a matrix B if either B = A or there exist matrices A0 , A1 , . . . , At (t > 1) such that A0 = A, At = B and Ar is a contraction of Ar−1 for r = 1, . . . , t. One can find the following fact in [2]: let A be a nonnegative integral matrix of order n > 1 and let B be a contraction of A. Then per A = per B.

(3)

2. k-Lucas numbers In this section, we determine a class of bipartite graphs whose number of 1-factors (k) . is ln−1 A matrix is said to be a (0,1)-matrix if each of its entries is either 0 or 1. Let k and n be positive integers. Let Tn = [tij ] be the n × n tridiagonal (0,1)-ma(k) trix such that tij = 1 if and only if |j − i| 6 1. Let Un = [uij ] be the n × n (0,1)matrix defined by uij = 1 if and only if 2 6 j − i 6 k − 1. It should be observed (k) (n) that Un = Un for k > n. In [4,5], the author found a class of bipartite graphs whose number of 1-factors is the nth k-Fibonacci number and the following result was proven: Theorem 2.1. Let G(F(n,k) ) be the bipartite graph with bipartite adjacency matrix (k) (k) F(n,k) = [fij ] = Tn + Un . Then the number of 1-factors of G(F(n,k) ) is gn+k−1 . Since the Fibonacci numbers are connected by the fundamental recursion Fn = Fn−1 + Fn−2 , it follows immediately that the Lucas numbers are likewise related by Ln = Ln−1 + Ln−2 for n > 2. We now consider the k-Lucas numbers. Lemma 2.2. For n > k, (k)

(k)

(k)

ln(k) = ln−1 + ln−2 + · · · + ln−k . (k)

Proof. Since, by definition of ln , (k)

(k)

ln−(k−1) =gn−(k−1)−1 + gn(k) .. . (k) (k) (k) =gn−2 + gn+k−2 ln−1

G.-Y. Lee / Linear Algebra and its Applications 320 (2000) 51–61 (k)

55

(k)

ln(k) =gn−1 + gn+k−1 . By applying (1), we have   (k)   (k) gn−(k−1)−1 + gn(k) ln−(k−1)   .   ..   .     .   . =     (k)   g (k) + g (k)  ln−1   n−2 n+k−2  (k)

ln

(k) (k) gn−1 + gn+k−1  (k) (k)  gn−(k−1)−2 + gn−1   ..     . =Qk    g (k) + g (k)   n−3 n+k−3  (k)

(k)

gn−2 + gn+k−2  (k)  ln−k  .   .   .  =Qk  . l (k)   n−2  (k)

ln−1 From (2), the proof is completed.  Let Sn(k) = [sij ] be the n × n (0,1)-matrix defined by sij = 1 if and only if −1 6 P (k) j − i 6 k − 1. For k < n, let C(n,k) = Sn − kj =2 E1j + E1k+1 , where Eij denotes the n × n matrix with 1 in the (i, j ) position and zeros elsewhere. If k 6 n, then P (k) the matrix E1j +1 , j > k, is not defined, and hence we let C(n,k) = Sn − nj=2 E1j for n 6 k. Then we have the following theorem. Theorem 2.3. Let G(C(n,2) ) be the bipartite graph with bipartite adjacency matrix C(n,2) , n > 3. Then the number of 1-factors of G(C(n,2) ) is Ln−1 . Proof. If n = 3, then we have C(3,2)

 1 = 1 0

0 1 1

 1 1 1

and hence per C(3,2) = 3 = L2 .

56

G.-Y. Lee / Linear Algebra and its Applications 320 (2000) 51–61 (n,2)

Let Cp

C(n,2)

be the pth contraction of C(n,2) , 1 6 p 6 n − 2. Since   1 0 1 0 ··· 0   1 1 1 0 · · · 0      0 1 1 1 · · · 0   = . . . , . . .  .. .. . . ..  .. ..      0 0 · · · 1 1 1   0 0 ··· 0 1 1

the matrix C(n,2) can be contracted on column 1 so that   1 2 0 0 ··· 0   1 1 1 0 · · · 0      0 1 1 1 · · · 0   (n,2) C1 = . . . . .  .. .. . . . . . . . . ..      0 0 · · · 1 1 1   0 0 ··· 0 1 1 (n,2)

Since the matrix C1 can be contracted on column 1 and L1 = 1, L2 = 3,   3 1 0 0 ··· 0   1 1 1 0 · · · 0     0 1 1 1 · · · 0   (n,2)   C2 =  . . . . . . .. .. . . ..    .. ..   0 0 · · · 1 1 1   0 0 ··· 0 1 1   L2 L1 0 0 ··· 0   1 1 1 0 · · · 0     1 1 1 · · · 0 0 = . ..  .. . .. .. ..  . . ..  . . .   0 0 ··· 1 1 1   0 0 ··· 0 1 1 (n,2)

Furthermore, the matrix C2

can be contracted on column 1 so that

G.-Y. Lee / Linear Algebra and its Applications 320 (2000) 51–61

(n,2)

C3

 L3  1   0 =  ..  .  0  0

L2

0

0

···

1

1

0

···

1 .. .

1 .. .

1 .. .

··· .. .

0

···

1

1

0

···

0

1

Continuing this process, we have  Lp Lp−1 0  1 1 1   1 1 0 Cp(n,2) =  ..  .. ..  . . .  0 0 ···  0 0 ··· for 3 6 p 6 n − 4. Hence,  Ln−3 Ln−4 (n,2) Cn−3 = 1 1 0 1

57

 0  0   0 ..  . .  1  1

0

···

0

···

1 .. .

··· .. .

1

1

0

1

0



 0   0 ..   .  1  1

 0 1 , 1

(n,2) on column 1, gives which, by contraction of Cn−4     Ln−2 Ln−3 Ln−3 + Ln−4 Ln−3 (n,2) = . Cn−2 = 1 1 1 1 (n,2) By applying (3), we have per C(n,2) = per Cn−2 = Ln−1 .



Let A(n) = [aij ] be the n × n (0,1)-matrix defined by aij = 1 if and only if −1 6 j − i. Brualdi [1] gave the following result: for n > 2, per A(n) = 2n−1 . From the above result, we have the following lemma. Lemma 2.4. For 2 6 n 6 k, (k) . per C(n,k) = 2n−2 = ln−1

Proof. Since n 6 k,

(4)

58

G.-Y. Lee / Linear Algebra and its Applications 320 (2000) 51–61

C(n,k)

 1 1   = 0  .. .

0 1 1 .. .

··· ··· ··· .. .

0 1 1 .. .

0

0

···

1

 0 1  1 . ..  . 1

Then = per A(n−1) . per C(n,k) = per C(n,k) 1 By (4), the lemma is proved.  In the following theorem, we have a class of bipartite graphs whose number of 1-factors equals the (n − 1)st k-Lucas number. Theorem 2.5. Let k and n be positive integers. Let G(C(n,k) ) be the bipartite graph with bipartite adjacency matrix C(n,k) , n > 3. Then, for n > k, the number of 1(k) factors of G(C(n,k) ) is the (n − 1)st k-Lucas number ln−1 . In particular, if n 6 k, (k)

then the number of 1-factors equals ln−1 = 2n−2 .

Proof. If n 6 k, then we are done, by Lemma 2.4. If k = 2, then we are also done, by Lemma 2.3. (p) (n,k) Now, suppose that n > k > 3. Let Cp = [cij ] be the pth contraction of C(n,k) = [cij ] for 1 6 p 6 n − 2. We only consider the first row because the entries of other (n,k) rows of Cp are 0 or 1. For q > k + 1, we can easily verify that if p = 1, then (1) (1) = · · · = c1k−1 = l1(k) = 1, c11

(1) c1k = l2(k) = 2,

(1) c1q = 0.

And, for q > k + 1, if p = 2, then (2)

(2)

(k)

(2)

c11 = · · · = c1k−2 = l2 = 2, (2)

(k)

c1k = l1 = 1,

(k)

(k)

c1k−1 = l1 + l2 = 1 + 2,

(2)

c1q = 0.

By repeated contractions, for p 6 k − 1, we have (p)

(p)

c11 = · · · = c1k−p = 2p−1 , c1k−p+s = 2p−1 − 2s−2 ,

c1k−p+1 = 2p−1 + 1,

2 6 s 6 p.

Since (k)

lj = 2j −1 , j = 1, 2, . . . , k − 1

(k)

and lk = 1 + 2k−1 for q > k + 1,

we have (k−1) (k) c11 =2k−2 = lk−1 ,

(k−1) (k) c12 = 1 + 2k−2 = l1(k) + lk−1 ,

G.-Y. Lee / Linear Algebra and its Applications 320 (2000) 51–61 (k−1) c13 =2k−2 − 1 =

k−2 X

(k−1) c14 = 2k−2 − 2 =

lq(k) ,

q=1

k−2 X

59

lq(k) ,

q=2

.. . (k−1) =2k−2 − 2k−4 = c1k−1

k−2 X

(k−1) (k) lq(k) , c1k = 2k−2 − 2k−3 = lk−2 ,

q=k−3 (k−1) c1q = 0.

That is, the matrix C(n,k) k−1 is as follows:  (k) Pk−2 (k) Pk−2 (k) (k) (k) (k) lk−1 l1 + lk−1 · · · lk−2 q=1 lq q=2 lq  1 1 1 1 ··· 1   0 1 1 1 · ·· 1   . . . . . .. .. .. .. ..  .. . 0 0 0 0 ··· 0 By the contraction of (p)

c11 = lp(k) ,

(p)

(n,k) Ck−1 for p−1 X

 0 0  0 . ..  . 1

k 6 p 6 n − k, we have

c12 =

lj(k) ,

(p)

c13 =

j =p−k+1 (p) and c1q

0 ··· 0 1 ··· 0 1 ··· 0 .. .. .. . . . 0 ··· 1

p−1 X

(p)

(k) lj(k) , . . . , c1k = lp−1 ,

j =p−k+2

= 0 for q > k + 1. That is, the

(n,k) matrix Cp , k

6 p 6 n − k, is the follow-

ing:  (k) Pp−1 (k) Pp−1 (k) (k) · · · lp−1 lp q=p−k+1 lq q=p−k+2 lq  1 1 1 ··· 1   0 1 1 ··· 1   . . . .. .. .. ..  .. . . 0 0 0 ··· 0

0 ··· 0 1 ··· 0 1 ··· 0 .. .. .. . . . 0 ··· 1

 0 0  0 . ..  .

(5)

1

Since p 6 n − 2, p − k + 1 6 n − k − 1. By the definition of C(n,k) = [cij ], we know that cin = 0 for i = 1, 2, . . . , n − k and cin = 1 for i = n − k + 1, . . . , n. From (5), by repeated contractions, we have (p) c11

=

lp(k) ,

(p) c12

=

p−1 X

(k)

lj ,

j =p−k+1 (p)

c13 =

p−1 X

(k)

(p)

lj , . . . , c1n−p =

j =p−k+2

Thus, for n − k + 1 6 p, we have

p−1 X j =n−k−1

(k)

lj .

60

G.-Y. Lee / Linear Algebra and its Applications 320 (2000) 51–61  (k) Pp−1 (k) Pp−1 (k) lp q=p−k+1 lq q=p−k+2 lq  1 1  1  1 1  0 C(n,k) = p   .. .. ..  . . . 0 0 0

··· ··· ··· .. . ···

Pp−1

(k) q=n−k−2 lq

(k)  q=n−k−1 lq

Pp−1

1 1 .. . 1

1 1 .. . 1

   .   

By repeated contractions, we have  (k)  (k) (k) ln−2 ln−3 + · · · + ln−k−1 (n,k) per Cn−2 =per 1 1 (k)

(k)

(k)

=ln−2 + ln−3 + · · · + ln−k−1 . By Lemma 2.2, the proof is completed.  Let B (n) = Tn + E13 − E23 + E24 − E34 and let G(B (n) ) be the bipartite graph with bipartite adjacency matrix B (n) . Lemma 2.6. Let G(B (n) ) be the bipartite graph with bipartite adjacency matrix B (n) . Then the bipartite graph G(B (n) ) is not isomorphic to the bipartite graph G(C(n,2) ). Proof. From the definition of C(n,k) , we have C(n,2) = Sn(2) − E12 + E13 . If k = 2, (2) then Sn = Tn and hence C(n,2) = Tn − E12 + E13 . Then the third column of C(n,2) has exactly four ones. But the matrix B (n) has no column that has exactly four ones. That is, there are no permutation matrices P and Q such that P B (n) Q = C(n,2) . Thus, the bipartite graph G(B (n) ) is not isomorphic to G(C(n,2) ).  The next theorem shows that we can find a bipartite graph whose number of 1-factors is the (n − 1)th Lucas number Ln−1 . Theorem 2.7. For n > 4, let G(B (n) ) be a bipartite graph with bipartite adjacency matrix B (n) . Then the number of 1-factors of G(B (n) ) is the (n − 1)th Lucas number Ln−1 . Proof. If n = 4, then  1 1 1 1 1 0 per  0 1 1 0 0 1

 0 1 =4 0 1

and L3 = 4. By induction on n, we assume that

G.-Y. Lee / Linear Algebra and its Applications 320 (2000) 51–61

61

per B (n) = Ln−1 and we consider n + 1. Let D be an n × 1 matrix such that   0 ..    D = .  . 0 1 Then B (n+1) =



B (n) DT

 D . 1

Let Bp(n) be the pth contraction of B (n) . Since   Ln−2 Ln−3 (n) Bn−2 = , 1 1 we have

 Ln−2 = per  1 per B (n+1) =per 0 =Ln−1 + Ln−2 = Ln . (n+1) Bn−2

Therefore, per B (n) = Ln−1 .

Ln−3 1 1

 0 1 1



Theorem 2.7 gives us a class of bipartite graphs that are not isomorphic to the bipartite graph G(C(n,2) ).

References [1] R.A. Brualdi, An interesting face of the polytope of doubly stochastic matrices, Linear Algebra Appl. 17 (1985) 5–18. [2] R.A. Brualdi, P.M. Gibson, Convex polyhedra of doubly stochastic matrices I: applications of the permanent function, J. Combin. Theory A22 (1977) 194–230. [3] R. Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985. [4] G.Y. Lee, S.G. Lee, A note on generalized Fibonacci numbers, The Fibonacci Q. 33 (3) (1995) 273– 278. [5] G.Y. Lee, S.G. Lee, H.G. Shin, On the k-generalized Fibonacci matrix Qk , Linear Algebra Appl. 251 (1997) 73–88. [6] E.P. Miles Jr., Generalized Fibonacci numbers and associated matrices, Amer. Math. Monthly (1960) 745–752. [7] H. Minc, Permanents, Encyclopedia of Mathematics and its Applications, Addison-Wesley, New York, 1978.