On the Number of 1-factors of Bipartite Graphs

Transcription

1 Math Sci Lett 2 No (2013) 181 Mathematical Scieces Letters A Iteratioal Joural O the Number of 1-factors of Bipartite Graphs Mehmet Akbulak 1 ad Ahmet Öteleş 2 1 Departmet of Mathematics Art ad Sciece Faculty Siirt Uiversity TR Siirt Turkey 2 Departmet of Mathematics Educatio Faculty Dicle Uiversity TR Diyarbakir Turkey Received: 28 Feb 2013 Revised: 2 May 2013 Accepted: 5 May 2013 Published olie: 1 Sep 2013 Abstract: I this paper we ivestigated relatioships betwee the Fiboacci Lucas Padova umbers ad 1-factors of some bipartite graphs with upper Hesseberg adjacecy matrix We calculated permaet of these upper Hesseberg matrices by cotractio method ad show that their permaets are equal to elemets of the Fiboacci Lucas ad Padova umbers At the ed of the paper we give some Maple 13 procedure i order to calculate umbers of 1-factors of above-metioed bipartite graphs Keywords: Fiboacci Sequece Lucas Sequece Padova Sequece 1-Factor Permaet 1 Itroductio The well-kow Fiboacci sequece {F } is defied by the recurrece relatio for >2 F F 1 + F 2 where F 1 F 2 1 The well-kow Lucas sequece {L } is defied by the recurrece relatio for >2 L L 1 + L 2 where L 1 1L 2 3 The Padova sequece {P } is defied by the recurrece relatio for >2 P P 2 + P 3 where P 0 P 1 P 2 1 [1] The first few values of these sequeces are give below: F L P where the summatio exteds over all permutatios σ of the symmetric group S Let A[a i j ] be a m real matrix with row vectors r 1 r 2 r m We say A is cotractible o colum (resp row) k if colum (resp row) k cotais exactly two ozero etries Suppose A is cotractible o colum k with a ik 0 a jk ad i j The the (m 1) ( 1) matrix A i j:k obtaied from A by replacig row i with a jk r i + a ik r j ad deletig row j ad colum k is called the cotractio of A o colum k relative to rows i ad j If A is cotractible o row k with a ki 0 a k j ad i j the [ T the matrix A k:i j A T i j:k] is called the cotractio of A o row k relative to colums i ad j We say that A ca be cotracted to a matrix B if either B A or there exist matrices A 0 A 1 A t (t 1) such that A 0 A A t B ad A r is a cotractio of A r 1 for r 1t Oe ca fid the followig fact i [2]: Let A be a oegative itegral matrix of order for > 1 ad let B be a cotractio of A The pera perb (1) The permaet of a -square matrix A [a i j ] is defied by pera a iσ(i) σεs i1 It is kow that there are a lot of relatios betwee permaets of matrices ad well-kow umber sequeces For example i [3] Mic defies geeralized Correspodig author makbulak@gmailcom Natural Scieces Publishig Cor

2 182 M Akbulak Ahmet Öteleş: O the Number of 1-factors of Bipartite Graphs Fiboacci umbers of order r as f (r) r k1 0 if <0 1 if 0 f ( kr) if >0 Mic also defies the super-diagoal (0 1) matrix F(r)[ f i j ] where { 1 if 1 j i r 2 f i j 0 otherwise ad proves that per(f(r)) f (r 1) I [4] the authors deote the matrix F(r) i [3] as F (k) ad obtai permaet of this matrix the same result i [3 Theorem 2] by applyig cotractio to the matrix F (k) I [5] Kilic defies the super-diagoal (012)- matrix S(k) as: S(k) ad proves that the permaet of S(k) equals to the (+1)th geeralized k-pell umber I [6] authors defie the square(01) matrix as H() ad by usig cotractio method they obtai perh() P 2 where P is the th Padova umber A bipartite graph G is a graph whose vertex set V ca be partitioed ito two two subsets V 1 ad V 2 such that every edge of G jois a vertex i V 1 ad a vertex i V 2 A 1 factor (or perfect matchig) of a graph with 2 vertices is a spaig subgraph of G i which every vertex has degree 1 The eumeratio or actual costuctio of 1 factors of a bipartite graph has may applicatios for example i maximal flow problems ad i assigmet ad schedulig problems Let A(G) be adjacecy matrix of the bipartite graph G ad let µ(g) deote the umber of 1 factors of G The oe ca fid the followig fact i [7]: µ(g) pera(g) Also oe ca fid more applicatios of permaets i [7] Let G be a bipartite graph whose vertex set V is partitioed ito two subsets V 1 ad V 2 such that V 1 V 2 We costruct the bipartite adjacet matrix B(G) [b i j ] of G as followig: b i j 1 if ad oly if G cotais a edge from v i V 1 to v j V 2 ad otherwise The i [7] the umber of 1 factors of bipartite graph G equals the permaet of its bipartite adjacey matrix I [8] Lee defies bipartite adjacecy matrix of bipartite graph G(L (k) ) the followig way: Let S (k) [s i j ] be the (01) matrix defied by s i j 1 if ad oly if 1 j i k 1 For k < L (k) S (k) k j2 E 1 j + E 1k+1 where E i j deotes the matrix with 1 i the (i j) positio ad zeros elsewhere Clearly (k) L ad he also shows that the umbers of 1 factors of G(L (2) ) ad G(L (k) ) is L 1 ad l (k) 1 where L ad l (k) are th Lucas ad k Lucas umbers respectively I [9] the authors defie bipartite adjacecy matrices of G(V ) ad G(W ) bipartite graphs as follows: V W ad they show that the umbers of 1 factors of G(V ) ad G(W ) are i0 F i ad i0 2 L i respectively I other words perv i0 F i perw i0 2 L i I [10] the authors defie families of square matrices such that (i) each matrix is the adjacecy matrix of a bipartite graph; ad (ii) the permaet of the matrices are the geeralized order k Lucas umbers ad a sum of cosecutive geeralized order k Fiboacci or Lucas umbers I this paper we give families of (01) upper Hesseberg matrices such that each of these matrices is the bipartite adjacet matrix correspodig to a bipartite graph ad the we show that the permaets of these Natural Scieces Publishig Cor

3 Math Sci Lett 2 No (2013) / wwwaturalspublishigcom/jouralsasp 183 matrices are equal to the well-kow Fiboacci Lucas ad Padova sequeces At the ed of this paper we give some Maple 13 procedures i order to calculate the umbers of 1-factors of bipartite graphs metioed above 2 Mai Results I this sectio we cosider a class of bipartite graphs The we show that the umbers of 1-factors of the graphs equal to Fiboacci Lucas ad Padova umbers Let U [u i j ] be the square (01) upper Hesseberg matrix defied by U (2) where 1 if j i 1 u i j 1 if i j ad j i 0 (mod2) 0 otherwise Theorem 21 Let G(U ) be the bipartite graph with bipartite adjacecy matrix U as i (2) for 3 The the umber of 1 factors of G(U ) is F Proof If 3 the we get peru F 3 (3) 0 Let U p be the pth cotractio of U 1 p 2 Sice the matrix U ca be cotracted o colum 1 we get 1 U 1 Sice the matrix U 1 ca be cotracted o colum 1 F 2 1 ad F 3 2 we write U 2 F 3 F 2 F 3 F 2 F 3 F 2 F 3 Furthermore if the matrix U 2 ca be cotracted o colum 1 we write F 4 F 3 F 4 F 3 F 4 F 3 F 4 U 3 Cotiuig this process we get F p+1 F p F p+1 F p F p+1 F p F p+1 U p for 3 p 4 Hece U 3 which by cotractio of U 3 U 2 F 2 F 3 F [ F 2 + F 3 F 2 o colum 1 gives ] [ ] F 1 F 2 By applyig (1) we get peru peru ( 2) F ad the proof is completed Example 21 Let the matrix U 3 as i (3) be bipartite adjacecy matrix of the graph G(U 3 ) The bipartite Natural Scieces Publishig Cor

4 184 M Akbulak Ahmet Öteleş: O the Number of 1-factors of Bipartite Graphs graph G(U 3 ) ca be see as: ad its 1-factors ca be see as: Let V [v i j ] be the square (01) upper Hesseberg matrix defied as the followig form: V 0 0 U ( 2) (4) 0 0 Theorem 22 Let G(V ) be the bipartite graph with bipartite adjacecy matrix V as i (4) for 3 The the umber of 1 factors of G(V ) is L 1 Proof If 3 the we get perv L 2 (5) 0 Let V r be the rth cotractio of V for 1 r 2 Sice the matrix V ca be cotracted o colum 1 we get V Sice the matrix V 1 ca be cotracted o colum 1 L 1 1 ad L 2 3 we write V 2 L 2 L 1 L 2 L 1 L 2 L 1 L Furthermore if the matrix V 2 ca be cotracted o colum 1 we write L 3 L 2 L 3 L 2 L 3 L 2 L 3 V 3 Cotiuig this process we get L r L r 1 L r L r 1 L r L r 1 L r V r for 3 r 4 Hece V 3 which by cotractio of V 3 V 2 L 3 L 4 L [ L 3 + L 4 L 3 o colum 1 gives ] [ ] L 2 L 3 By applyig (1) we get perv perv ( 2) L 1 ad the proof is completed Natural Scieces Publishig Cor

5 Math Sci Lett 2 No (2013) / wwwaturalspublishigcom/jouralsasp 185 Example 22 Let the matrix V 3 as i (5) be bipartite adjacecy matrix of the graph G(V 3 ) The bipartite graph G(V 3 ) ca be see as: ad its 1-factors ca be give as: Let W [w i j ] be the square (01) upper Hesseberg matrix defied by 1 W (6) where 1 if i1 1 if j i 1 w i j 1 if j i ad j i 1 (mod 3) 0 otherwise Theorem 23 Let G(W ) be the bipartite graph with bipartite adjacecy matrix W as i (6) for 3 The the umber of 1 factors of G(W ) is P Proof If 5 the we get perw P 5 (7) Let W t be the tth cotractio of W 1 t 2 Sice the matrix W ca be cotracted o colum 1 we get W Sice the matrix W 1 ca be cotracted o colum 1 we write W Furthermore sice the matrix W 2 ca be cotracted o colum 1 P 3 P 4 2 ad P 5 3 we ca write W P 4 P 5 P 3 P 4 P 5 P 3 P 4 P 5 P Cotiuig this process we get P r+1 P r+2 P r P r+1 P r+2 P r P r+1 P r+2 P r W t for 3 r 4 Hece W 3 P 2 P 1 P Natural Scieces Publishig Cor

6 186 M Akbulak Ahmet Öteleş: O the Number of 1-factors of Bipartite Graphs which by cotractio of W 3 W 2 [ P 1 P 2 + P o colum 1 gives ] [ ] P 1 P 1 0 By applyig (1) we get perw perw ( 2) P ad the proof is completed Example 23 Let the matrix W 3 as i (7) be bipartite adjacecy matrix of the graph G(W 3 ) The bipartite graph G(W 3 ) ca be see as: ad its 1-factors ca be give as: 3 Coclusio We showed that umbers of 1-factors of some bipartite graphs are equal to the well-kow Fiboacci Lucas ad Padova umbers Appedix A The followig procedure calculates the umber of 1 factors of bipartite graph G(U ) give i Theorem 1 restart: with(liearalgebra): permaet:proc() local ijpuu; u:(ij)->piecewise(j>i ad j-i mod 2 01i-j110); U:Matrix(u): for p from 0 to -2 do prit(pu): for j from 2 to -p do U[1j]:U[21]*U[1j]+U[11]*U[2j]: U:DeleteRow(DeleteColum(Matrix(-ppU)1)2): prit(peval(u)): ed proc:with(liearalgebra): permaet( ); Appedix B The followig procedure calculates the umber of 1 factors of bipartite graph G(V ) give i Theorem 2 restart: with(liearalgebra): permaet:proc() local ijrvv; v:(ij)->piecewise(i11i2 ad j mod 2 11i>2 ad j>i ad j-i mod 2 01i-j110); V:Matrix(v): for r from 0 to -2 do prit(rv): for j from 2 to -r do V[1j]:V[21]*V[1j]+V[11]*V[2j]: V:DeleteRow(DeleteColum(Matrix(-rrV)1)2): prit(reval(v)): ed proc:with(liearalgebra): permaet( ); Appedix C The followig procedure calculates the umber of 1 factors of bipartite graph G(W ) give i Theorem 3 restart: with(liearalgebra): permaet:proc() local ijtww; w:(ij)- >piecewise(i11j>i ad j-i mod 3 11ij110); W:Matrix(w): for t from 0 to -2 do prit(tw): for j from 2 to -t do W[1j]:W[21]*W[1j]+W[11]*W[2j]: W:DeleteRow(DeleteColum(Matrix(-ttW)1)2): prit(teval(w)): ed proc:with(liearalgebra): permaet( ); Natural Scieces Publishig Cor

7 Math Sci Lett 2 No (2013) / wwwaturalspublishigcom/jouralsasp 187 Refereces [1] A G Shao P G Aderso A F Horadam Properties of Cordoier Perri ad Va der Laa umbers Iteratioal Joural of Mathematical Educatio i Sciece ad Techology (2006) [2] R A Brualdi PM Gibso Covex polyhedra of doubly stochastic matrices I: applicatios of the permaet fuctio J Combi Theory A (1977) [3] H Mic Permaets of (01)-Circulats Caad Math Bull (1964) [4] G Y Lee S G Lee A ote o geeralized Fiboacci umbers The Fiboacci Q (1995) [5] E Kilic O the usual Fiboacci ad geeralized order-k Pell umbers Ars Combiatoria (2008) [6] F Yilmaz D Bozkurt Some properties of Padova sequece by matrix methods Ars Combiatoria (2012) [7] H Mic Permaets Ecyclopedia of mathematics ad its applicatios Addiso-Wesley New York (1978) [8] G Y Lee k-lucas umbers ad associated bipartite graphs Liear Algebra Appl (2000) [9] E Kilic D Tasci O families of bipartite graphs associated with sums of Fiboacci ad Lucas umbers Ars Combiatoria (2008) [10] E Kilic D Tasci O families of bipartite graphs associated with sums of geeralized order-k Fiboacci ad Lucas umbers Ars Combiatoria (2008) Natural Scieces Publishig Cor