ON THE NUMBER OF LAPLACIAN EIGENVALUES OF TREES SMALLER THAN TWO. Lingling Zhou, Bo Zhou* and Zhibin Du 1. INTRODUCTION

Transcription

1 TAIWANESE JOURNAL OF MATHEMATICS Vol 19, No 1, pp 65-75, February 015 DOI: /tjm This paper is available olie at ON THE NUMBER OF LAPLACIAN EIGENVALUES OF TREES SMALLER THAN TWO Liglig Zhou, Bo Zhou* ad Zhibi Du Abstract Let m T [0, ) be the umber of Laplacia eigevalues of a tree T i [0, ), multiplicities icluded We give best possible upper bouds for m T [0, ) usig the parameters such as the umber of pedat vertices, diameter, matchig umber, ad domiatio umber, ad characterize the trees T of order with m T [0, ) = 1,, ad, respectively, ad i particular, show that m T [0, ) = if ad oly if the matchig umber of T is 1 INTRODUCTION We cosider simple graphs Let G be a graph with vertex set V (G) For v V (G), letd G (v) bethedegreeofv i G The Laplacia matrix of G is defied as L(G) =D(G) A(G), whered(g) is the degree diagoal matrix of G, ada(g) is the adjacecy matrix of G The Laplacia eigevalues of G are the eigevalues of L(G) SiceL(G) is a positive semi-defiite matrix, the Laplacia eigevalues of G are oegative real umbers Let μ 1 (G) μ (G) μ (G) be the Laplacia eigevalues of G, arraged i odecreasig order, where = V (G) Sice each row sum of L(G) is zero, μ 1 (G) =0 Recall that μ (G) (see [1, 5]) Thus all Laplacia eigevalues of G belog to [0,] For a survey o Laplacia eigevalues, see [11] For a graph G o vertices ad a iterval I [0,], letm G I be the umber of Laplacia eigevalues of G, multiplicities icluded, that belog to I Groe ad Merris [5] showed that for a graph with at least oe edge, its largest Laplacia eigevalue is at least the maximum degree plus oe Thus for a tree T o vertices, m T [0, ) 1 Received February 8, 014, accepted April 8, 014 Commuicated by Gerard Jehwa Chag 010 Mathematics Subject Classificatio: 05C50, 05C35 Key words ad phrases: Laplacia eigevalues, Trees, Pedat vertex, Diameter, Matchig umber, Domiatio umber *Correspodig author 65

2 66 Liglig Zhou, Bo Zhou ad Zhibi Du A vertex of a graph G is a pedat vertex if d G (v) =1 A vertex of G is a quasi-pedat vertex if it is adjacet to a pedat vertex For a graph G o vertices with p pedat vertices, q quasi-pedat vertices, ad diameter d, Groeet al [6] showed that m G [0, 1],m G [1,] p, m G [0, 1),m G (1,] q, d m G (,], ad Merris [10] showed that if >q, the m G (,] q Braga et al [3] showed that for a tree T o vertices, m T [0, ) More results alog this lie may be foud i [3, 7, 8] I this paper, we give best possible upper bouds for m T [0, ) usig the parameters of a tree T such as the umber of pedat vertices, diameter, matchig umber, ad domiatio umber, provide a simple differet proof for the lower boud i [3] metioed above, characterize the trees T of order with m T [0, ) = 1,, ad, respectively, ad i particular, show that mt [0, ) = if ad oly if the matchig umber of T is (i Theorem 4) PRELIMINARIES A algorithm for computig the umber of Laplacia eigevalues of a tree i a iterval was proposed i [3] based o the algorithm for computig the umber of adjacecy eigevalues of a tree i a iterval [9] For a tree T o vertices, choose ay vertex as the root of T, ad label the vertices of T as v 1,v,,v such that if v i is a child of v k,thek>i The algorithm for computig m T [0, ) of a tree T is give as follows: Iput: tree T Output: diagoal matrix D cogruet to L(T ) Algorithm Diagoalize L(T ) iitialize a T (v) :=d T (v) for all vertices v order vertices bottom up for k =1to if v k is a leaf the cotiue

3 O the Number of Laplacia Eigevalues of Trees Smaller tha Two 67 else if a T (c) 0for all childre c of v k the a T (v k ):=a T (v k ) 1 c is a child of v k a T (c) else select oe child v j of v k for which a T (v j )=0 a T (v k ):= 1 a T (v j ):= if v k has a paret v l, the remove the edge v k v l ed loop For a tree T with vertices v 1,v,,v labelled as above, the weight of v i i T is the i-th diagoal etry a T (v i ) of the diagoal matrix D obtaied uder the above algorithm, where 1 i Ifa T (v i ) < 0, wesayv i has a egative weight i T Lemma 1 [3] Suppose that T is a tree The m T [0, ) is equal to the umber of vertices with egative weights i T A double broom is a tree obtaied by attachig some pedat vertices to the two ed vertices of a path o at least two vertices I particular, a star is also regarded as a double broom Lemma Let T be a -vertex double broom with diameter d, where1 d 1 Them T [0, ) = d Proof Choosig a quasi-pedat vertex of T as the root of T The the result follows from Lemma 1 easily Lemma 3 Let T be a tree with v V (T ), ad T be the tree obtaied from T by attachig a path o two vertices to v Them T [0, ) = m T [0, ) + 1 Proof I both T ad T, we choose v as the root Note that the two vertices i T ot i T have weights 1 ad 1, ada T (x) =a T (x) for x V (T ) The the result follows from Lemma 1 clearly Lemma 4 Let T be a tree with v V (T ), ad T be the tree obtaied from T by attachig two pedat vertices to v Them T [0, ) m T [0, ) + 1 Proof Let us choose v as the root of both T ad T Clearly, a T (x) =a T (x) for x V (T ) \{v} Deote by s the umber of vertices i T differet from v with egative weights Note that each pedat vertex i T has weight 1 By Lemma 1, m T [0, ) s +=(s +1)+1 m T [0, ) + 1 Lemma 5 [6] LetG be a -vertex graph ad G a graph obtaied from G by deletig a edge The 0=μ 1 (G )=μ 1 (G) μ (G ) μ (G) μ (G ) μ (G)

4 68 Liglig Zhou, Bo Zhou ad Zhibi Du For a vertex v of a graph G, G v deotes the graph resultig from G by deletig v (ad its icidet edges) For a edge uv of a graph G (the complemet of G, respectively), G uv (G + uv, respectively) deotes the graph resultig from G by deletig (addig, respectively) uv 3 UPPER BOUNDS FOR m T [0, ) For a tree T,ifv is a vertex of T with exactly d T (v) 1 1 pedat eighbors, the the subgraph iduced by v ad its d T (v) 1 pedat eighbors is said to be a pedat star of T at v IfT is ot a star, the T has some pedat stars Lemma 31 Suppose that T is a tree with a pedat star at v, sayt 1 If we choose a vertex of T outside T 1 as the root of T,thea T (v) > 0 Proof Clearly, a T (u) = 1 for ay pedat eighbor u of v i T Thus 1 a T (v)=d T (v) (d T (v) 1) a T (u) =d T (v) 3 > 0, as desired Lemma 3 Let T be a tree, ad T 1 be the tree obtaied from T by deletig a pedat vertex The m T [0, ) = m T1 [0, ) or m T1 [0, ) + 1 Proof Let v be a pedat vertex of T, beig adjacet to u By Lemma 5, μ i (T ) μ i+1 (T uv) μ i+1 (T ) for 1 i 1 Obviously, T uv cosists of T 1 ad a isolated vertex v Thus μ i+1 (T uv) =μ i (T 1 ) for 1 i 1 It follows that μ i (T ) μ i (T 1 ) μ i+1 (T ) for 1 i 1 From μ i (T ) μ i (T 1 ),wehave m T [0, ) m T1 [0, ), ad from μ i (T 1 ) μ i+1 (T ),wehavem T1 [0, ) m T [0, ) 1 Thus we have the desired result Theorem 31 Let T be a -vertex tree with p pedat vertices, where p 1 Them T [0, ) +p 1 Proof We prove the result by iductio o If =3,theT is a star with p =, ad by Lemma, m T [0, ) = +p 1 Suppose that the result holds for all trees o less tha 4 vertices with ay possible umber of pedat vertices Let T be a -vertex tree with p pedat vertices Let v be a ed vertex of a diametrical path of T,adu be the (uique) eighbor of v (o that diametrical path) Suppose first that u is of degree two Note that T v u has at most p pedat vertices Applyig the iductio hypothesis to T v u, wehavem T v u [0, ) The by Lemma 3, we have ( )+p 1

5 O the Number of Laplacia Eigevalues of Trees Smaller tha Two 69 ( ) + p 1 + p 1 m T [0, ) = m T v u [0, ) = Now suppose that u is of degree at least three Note that T v has p 1 pedat vertices Applyig the iductio hypothesis to T v, wehavem T v [0, ) ( 1)+(p 1) 1 The by Lemma 3, we have ( 1) + (p 1) 1 m T [0, ) m T v [0, ) = The result follows + p 1 Corollary 31 Let T be a -vertex tree with diameter d, where d 1 The m T [0, ) d Proof Deote by p the umber of pedat vertices i T Clearly, p d +1 The the result follows from Theorem 31 easily The upper bouds i Theorem 31 ad Corollary 31 are both tight sice they are attaied whe T is a -vertex double broom A matchig of a graph is a edge subset i which o pair shares a commo vertex The matchig umber β(g) of a graph G is the maximum cardiality of a matchig of G Theorem 3 Let T be a -vertex tree with matchig umber β, where1 β Them T [0, ) β Proof We prove the result by iductio o The case =3follows obviously from Lemma Suppose that the result holds for all trees o less tha 4 vertices with ay possible matchig umber Let T be a -vertex tree with matchig umber β Letv be a ed vertex of a diametrical path of T,adube the (uique) eighbor of v (o that diametrical path) Suppose first that u is of degree two Note that T v u has matchig umber β 1 Applyig the iductio hypothesis to T v u, wehavem T v u [0, ) ( ) (β 1) = β 1 Now it follows from Lemma 3 that m T [0, ) = m T v u [0, ) + 1 ( β 1) + 1 = β Now suppose that u is of degree at least three Note that T v has matchig umber β Applyig the iductio hypothesis to T v, wehavem T v [0, ) 1 β Now it follows from Lemma 3 that m T [0, ) m T v [0, ) + 1 ( 1 β)+1= β

6 70 Liglig Zhou, Bo Zhou ad Zhibi Du The result follows A domiatig set of a graph is a vertex subset whose closed eighborhood cotais all vertices of the graph The domiatio umber of a graph G is the miimum cardiality of a domiatig set of G A coverig of a graph G is a vertex subset K such that every edge of G has at least oe ed vertex i K Corollary 3 Let T be a -vertex tree with domiatio umber γ, where1 γ / Them T [0, ) γ Proof Deote by β the matchig umber of T ByKöig s theorem [], β is equal to the miimum cardiality of a coverig of G Note that a coverig of T is also a domiatig set of T Thus β γ The the result follows from Theorem 3 easily The upper bouds i Theorem 3 ad Corollary 3 are both tight sice they are attaied whe T is a -vertex tree obtaied by attachig some paths o two vertices to the cetral vertex of a star Recall that m T [0, ) 1 for ay tree T o vertices [5], (which also follows from Theorem 3) Let T 1 diameter three, where 4 Let T diameter four, where 5 be the set of -vertex trees (double brooms) with be the set of -vertex double brooms with Theorem 33 Let T be a tree o vertices (i) m T [0, ) = 1 for if ad oly if T = S (ii) m T [0, ) = for 4 if ad oly if T T 1 T Proof By Lemma, m T [0, ) = 1 if T = S,adm T [0, ) = if T T 1 T Suppose i the followig that T {S } T 1 T The 6 Let P = v 0 v 1 v d be a diametrical path of T Obviously, d 4 Let T 1 be the pedat star of T at v 1,adT be the pedat star of T at v d 1 If T 1 ad T are the oly two vertex-disjoit pedat stars i T,theT is a double broom with d 5, ad thus by Lemma, m T [0, ) 3 Suppose that there are at least three vertex-disjoit pedat stars i T Let T 3 be a pedat star i T differet from T 1 ad T If V (T )=V(T 1 ) V (T ) V (T 3 ),thet is the tree obtaied by attachig at least oe pedat vertex to each vertex of P 3, ad by choosig v as the root of T ad applyig Lemma 1, we have m T [0, ) = 3 Suppose that V (T ) V (T 1 ) V (T ) V (T 3 ) Let u beavertexit outside T 1,T,T 3 Choosig u as the root of T, ad by Lemma 31, each of T 1,T,T 3 has oe vertex which is ot of egative weight Thus, by Lemma 1, we have m T [0, ) 3 Now the result follows easily

7 O the Number of Laplacia Eigevalues of Trees Smaller tha Two 71 4 A LOWER BOUND FOR m T [0, ) For a tree T,ifu is a pedat vertex of T beig adjacet to a vertex v of degree two, the the subgraph of T iduced by u ad v is said to be a pedat P of T For a tree o at least three vertices, if there is o pedat P, the there are two pedat vertices sharig a commo eighbor Deletig a pedat P of a tree T is said to be a deletig pedat P operatio, ad deletig a pedat P of T or two pedat vertices of T sharig a commo eighbor is said to be a geeralized deletig pedat P operatio For a tree o vertices, we ca fially obtai P 1 for odd ad P for eve by a series of geeralized deletig pedat P operatios The followig result has bee obtaied by Braga et al [3] Here we preset a simple differet reasoig Theorem 41 Let T be a tree o vertices The m T [0, ) Proof By Lemmas 3 ad 4, each geeralized deletig pedat P operatio decreases the umber of Laplacia eigevalues i [0, ) by at least oe Thus, if is odd, the m T [0, ) m P1 [0, ) + 1 = +1,adif is eve, the m T [0, ) m P [0, ) + = Lemma 41 Let T be a tree with a diametrical path P = v 0 v 1 v d,whered 4, ad for some i with i d, v i is of degree three Let T = T v i v i+1 +v i v i+1, where v i is the pedat eighbor of v i outside P Them T [0, ) m T [0, ) Proof Let us choose v i as the root of both T ad T It is easily checked that a T (x) =a T (x) for x V (T ) \{v i,v i }, a T (v i )= 1, a T (v i )= 1 a T (v i 1 ) 1 a T (v i+1 ), a T (vi )= 1 a T (v i+1 ) = 1 a T (v i+1 ), 1 a T (v i )= a T (v i 1 ) 1 a T (vi ) = 1 a T (v i 1 ) + a T (v i+1 ) Deote by s the umber of vertices i T differet from v i,vi with egative weights By Lemma 1, m T [0, ) s +1ad m T [0, ) s + Suppose by cotradictio that m T [0, ) <m T [0, ) The s +1 m T [0, ) m T [0, ) 1 s +1, ad thus m T [0, ) = s+1 ad m T [0, ) = s+, implyig that a T (v i ) 0, a T (v i ) < 0, ada T (v i ) < 0 Froma T (v i ) < 0, wehavea T (v i+1 ) > 0, adthe

8 7 Liglig Zhou, Bo Zhou ad Zhibi Du a T (v i ) a T (v i )=a T (v i+1 )+ Thus a T (v i ) a T (v i ) 0, which is a cotradictio 1 a T (v i+1 ) 0 Attachig the path P to a vertex of a tree T is called addig a pedat P to T By Lemma 3, each operatio of addig a pedat P icreases the umber of Laplacia eigevalues i [0, ) by oe Theorem 4 Let T be a tree o vertices The m T [0, ) = if ad oly if β(t )= Proof If β(t )=, the by Theorem 3, we have m T [0, ) =, ad thus m T [0, ) = Suppose that m T [0, ) = We will prove that β(t )= Claim 1 T is a tree obtaiable from P if is eve ad from P 1 if is odd by sequetially addig pedat P s Applyig a series of deletig pedat P operatios from T, we may fially obtai atreet (1) without pedat P Let (1) = V (T (1) ) By Lemma 3, we have m T (1)[0, ) = (1) If (1) =1or, ie, T (1) = P 1 or P, the Claim 1 follows obviously I the followig, we will prove that (1) =1or Sice T (1) has o pedat P,wehave (1) 3,adif (1) =4, 5, thet (1) is a star, ad thus m T (1)[0, ) = (1) 1 (1), which is a cotradictio, implyig that (1) 4, 5 Suppose that (1) 6 Letd be the diameter of T (1),adletP = v 0 v 1 v d be a diametrical path of T (1) Note that both v 1 ad v d 1 are of degree at least three (sice T (1) has o pedat P ) If d =, 3, thet (1) is a double broom, by Theorem 33, m T (1)[0, ) (1) > (1), which is a cotradictio Thus d 4 Note that the deletio of edges i P from T (1) results i a forest with d +1 compoets, each of which cotais exactly oe vertex of P Amog such d +1 compoets, deote by T i the oe cotaiig v i,where0 i d Let T () be the tree obtaied from T (1) by a series of geeralized deletig pedat P operatios such that oe vertex of T i is left if V (T i ) is odd ad two vertices of T i are left if V (T i ) is eve for all i d Let () = V (T () ) Now by Lemmas 3, 4, ad 41, we have

9 O the Number of Laplacia Eigevalues of Trees Smaller tha Two 73 (1) () = m T (1)[0, ) m T ()[0, ) + (1) () () + (1) () (1) = Thus m T ()[0, ) = Note that P = v 0 v 1 v d is still a diametrical path of T (), v 1 ad v d 1 are both of degree at least three, ad the vertices v,v 3,,v d are all of degrees two or three This implies that the diameter, say d, oft () satisfies that 4 d () 3 If the vertices v,v 3,,v d i T () are all of degree two, the T () is a double broom, ad by Lemma, we have () = m T ()[0, ) = d () () ( () 3) = () +3, which is a cotradictio Suppose that there is a vertex v i of degree three i T (),where i d Deote by vi the pedat eighbor of v i i T () outside P LetT = T () v i v i+1 + vi v i+1 Note that T has oe less vertex of degree three tha T () By Lemma 41, we have m T ()[0, ) m T [0, ) Repeatig the trasformatio from T () to T, we ca fially get a double broom T with () vertices such that the degrees of v 1 ad v d 1 i T are the same as those i T (), the vertices v,v 3,,v d ad their pedat eighbors i T () are all of degree two i T,ad () = m T ()[0, ) m T [0, ) Notethat T has diameter at most () 3 (sice v 1 ad v d 1 are both of degree at least three) As above, we ca deduce a cotradictio Thus (1) =1or, ad Claim 1 follows Obviously, each operatio of addig a pedat P icreases the matchig umber by oe By Claim 1, β(t )=β(p )+ = if is eve, ad β(t )=β(p 1)+ 1 = 1 if is odd Thus β(t )= 5 REMARK Recall that for a tree T o vertices, m T [0, ) 1 Theorem 51 For positive itegers, k with ad k 1, there exists a tree T o vertices such that m T [0, ) = k

10 74 Liglig Zhou, Bo Zhou ad Zhibi Du Proof Observe that m Sk + [0, ) = k +1 Let T be the -vertex tree obtaied by attachig a path o k vertices to a vertex of S k + By Lemma 3, we have as desired m T [0, ) = m Sk + [0, ) + k = k, ACKNOWLEDGMENTS This work was supported by the Specialized Research Fud for the Doctoral Program of Higher Educatio of Chia (No ) ad the Natioal Natural Sciece Foudatio of Chia (No ) REFERENCES 1 W N Aderso Jr ad T D Morley, Eigevalues of the Laplacia of a graph, Liear Multiliear Algebra, 18 (1985), J A Body ad U S R Murty, Graph Theory with Applicatios, America Elsevier, New York, R O Braga, V M Rodrigues ad V Trevisa, O the distributio of Laplacia eigevalues of trees, Discrete Math, 313 (013), M Fiedler, Algebraic coectivity of graphs, Czechoslovak Math J, 3 (1973), R Groe ad R Merris, The Laplacia spectrum of a graph II, SIAM J Discrete Math, 7 (1994), R Groe, R Merris ad V S Suder, The Laplacia spectrum of a graph, SIAM J Matrix Aal Appl, 11 (1990), J Guo ad S Ta, A relatio betwee the matchig umber ad Laplacia spectrum of a graph, Liear Algebra Appl, 35 (001), J Guo, X Wu, J Zhag ad K Fag, O the distributio of Laplacia eigevalues of a graph, Acta Math Si (Egl Ser), 7 (011), D P Jacobs ad V Trevisa, Locatig the eigevalues of trees, Liear Algebra Appl, 434 (011), R Merris, The umber of eigevalues greater tha two i the Laplacia spectrum of a graph, Port Math, 48 (1991), R Merris, Laplacia matrices of graphs: A survey, Liear Algebra Appl, (1994),

11 O the Number of Laplacia Eigevalues of Trees Smaller tha Two 75 Liglig Zhou ad Bo Zhou Departmet of Mathematics South Chia Normal Uiversity Guagzhou P R Chia zhoubo@scueduc Zhibi Du School of Mathematics ad Iformatio Scieces Zhaoqig Uiversity Zhaoqig P R Chia