TAIWANESE JOURNAL OF MATHEMATICS Vol 19, No 1, pp 65-75, February 015 DOI: 1011650/tjm190154411 This paper is available olie at http://jouraltaiwamathsocorgtw 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
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
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)
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
O the Number of Laplacia Eigevalues of Trees Smaller tha Two 69 ( ) + p 1 + p 1 m T [0, ) = m T v u [0, ) + 1 +1= 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, ) + 1 +1= 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
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
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
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
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
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 01440711000) ad the Natioal Natural Sciece Foudatio of Chia (No 11071089) REFERENCES 1 W N Aderso Jr ad T D Morley, Eigevalues of the Laplacia of a graph, Liear Multiliear Algebra, 18 (1985), 141-145 J A Body ad U S R Murty, Graph Theory with Applicatios, America Elsevier, New York, 1976 3 R O Braga, V M Rodrigues ad V Trevisa, O the distributio of Laplacia eigevalues of trees, Discrete Math, 313 (013), 38-389 4 M Fiedler, Algebraic coectivity of graphs, Czechoslovak Math J, 3 (1973), 98-305 5 R Groe ad R Merris, The Laplacia spectrum of a graph II, SIAM J Discrete Math, 7 (1994), 1-9 6 R Groe, R Merris ad V S Suder, The Laplacia spectrum of a graph, SIAM J Matrix Aal Appl, 11 (1990), 18-38 7 J Guo ad S Ta, A relatio betwee the matchig umber ad Laplacia spectrum of a graph, Liear Algebra Appl, 35 (001), 71-74 8 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), 59-68 9 D P Jacobs ad V Trevisa, Locatig the eigevalues of trees, Liear Algebra Appl, 434 (011), 81-88 10 R Merris, The umber of eigevalues greater tha two i the Laplacia spectrum of a graph, Port Math, 48 (1991), 345-349 11 R Merris, Laplacia matrices of graphs: A survey, Liear Algebra Appl, 197-198 (1994), 143-176
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 510631 P R Chia E-mail: zhoubo@scueduc Zhibi Du School of Mathematics ad Iformatio Scieces Zhaoqig Uiversity Zhaoqig 56061 P R Chia