R E S E A R C H
Open Access
On the sum of the two largest Laplacian
eigenvalues of trees
Mei Guan
1, Mingqing Zhai
2*and Yongfeng Wu
3*Correspondence:
2School of Mathematical Science,
Chuzhou University, Chuzhou, 239012, China
Full list of author information is available at the end of the article
Abstract
ForS(T), the sum of the two largest Laplacian eigenvalues of a treeT, an upper bound is obtained. Moreover, among all trees withn≥4 vertices, the unique tree which attains the maximal value ofS(T) is determined.
MSC: 05C50
Keywords: upper bound; tree; Laplacian matrix
1 Introduction
LetV(G) be the vertex set andE(G) be the edge set of a graphG. The numbers of ver-tices and edges ofGare denoted byn(G) andm(G), respectively. For a vertexv∈V(G), letNG(v) be the set of vertices adjacent tovanddG(v) =|NG(v)|be the degree ofv.
Partic-ularly, denote by(G) the maximum degree ofG. The diameter of a connected graphG, denoted byd(G), is the maximum distance among all pairs of vertices inG. LetA(G) be the adjacency matrix ofGandD(G) be the diagonal matrix of vertex degrees. The matrix
D(G) –A(G) is called theLaplacian matrixofGand its eigenvalues are called the Lapla-cian eigenvalues ofG. Letμ≥μ≥ · · · ≥μnbe the Laplacian eigenvalues of a graphG
withnvertices. It is well known thatμn= and
n–
i= μi= m(G). In particular,μn–is called the algebraic connectivity ofGand it is denoted byα(G).
The Laplacian matrix is an important topic in the theory of graph spectra. Particularly, much literature has paid attention toμ,μ,μn–orμ–μn–for trees (see, for example, [–]). LetSnbe the star of ordern,Ska,bbe the tree obtained from two starsSa+,Sb+by joining a path of lengthkbetween their central vertices (see Figure ). As is well known, among all trees of ordern,Snhas the largest value ofμ(see []) andSn–,has the second largest value ofμ(see []). On the other hand, Guo [] proved that these two trees also attain the first two smallest values ofμ, respectively. This implies thatμ,μcannot attain simultaneously the maximal (or minimal) value and even the relation between them seems like a seesaw. Therefore, it is interesting to investigate the value ofμ+μ. Moreover, Zhang [] showed that theS
k–,k–,Sk–,k–,Sk–,k–attain simultaneously the largest value ofμamong all trees with kvertices. Then Shaoet al.[] showed thatSk–,k–attains the largest value ofμamong all trees with k+ vertices.
Another motivation to study the value ofμ+μcame from a result of Haemerset al.[], who showed thatμ+μ≤m(G) + for any graphG. This result implies that Brouwer’s
Figure 1 Sk
a,b: a tree of ordera+b+k+ 1.
conjecture [],
μ+μ+· · ·+μk≤m(G) +
k+
,
is true fork= . Considering a treeT, we haveμ+μ≤n(T) + . Recently, Fritscheret
al.[] improved this bound by givingμ+μ<n(T) + –n(T). This paper determines the extremal tree that attains the bound ofμ+μ. Moreover, for general connected graphs, we also give a conjecture on the extremal graphs forμ+μ.
2 A sharp upper bound of
μ
1+μ
2LetSk(G) be the sum of the largestkLaplacian eigenvalues of a graphG. Whenk= , we
shall writeS(G) instead ofSk(G) for simplicity. For graphsGandH, we denote byG∪H
the graph with vertex setV(G)∪V(H) and edge setE(G)∪E(H). The following lemmas come from an important result as regards a real symmetric matrix.
Lemma . ([]) Let G,G, . . . ,Gr be some edge-disjoint graphs. Then Sk(
r i=Gi)≤
r
i=Sk(Gi)for any k.
Lemma .([]) For any graph G,S(G)≤m(G) + .
Lemma . Let G be a connected graph,di=dG(vi)and mi=vj∈NG(vi)dj/di.Then (i) []μ(G)≥(G) + ,with equality if and only if(G) =n(G) – .
(ii) []μ(G)≤n(G),with equality if and only if the complement ofGis disconnected. (iii) []μ(G)≤max{di+mi|vi∈V(G)}.
Lemma .([]) Let T be a tree of order n.If TSn,thenμ(T)≤μ(Sn–,),with equality
if and only if T∼=S
n–,.
Corollary . Let T be a tree with n vertices and diameter d≥.Thenμ(T) <n– ..
Proof Note that any treeThas diameterd≥ ifTSn. According to Lemma .,μ(T)≤ μ(Sn–,). Further, by Lemma .,
μ
Sn–,≤max{di+mi}=n– +
n–
n– =n– +
n– <n– .
forn≥. Forn= , a straightforward calculation shows thatμ(S,) = +√ < ..
Lemma . ([]) Let T be a tree of order n and diameter d ≥ . Then α(T) ≥
α(Sd– n–d+
,n–d+
),with equality if and only if T∼=Sd– n–d+
,n–d+
Lemma .([]) Let G be a graph with a vertex u of degree one.Thenα(G)≤α(G–u).
Lemma . implies that the algebraic connectivity of a tree is not greater than that of its subtree.
Lemma .([]) Let Tk
n(n≥k+ )be a tree obtained from a star Sn–kby replacing its k
edges with k paths of length two,respectively.If k≥,thenμ(Tnk) =+
√ .
The following lemma can be found in [] and is known as the Interlacing Theorem of Laplacian eigenvalues.
Lemma . Let G be a graph of order n and H be a graph obtained from G by deleting an edge.Then
μ(G)≥μ(H)≥ · · · ≥μn(G)≥μn(H) = .
Next we give the main theorem of this section. Its proof is divided into several sequent claims.
Theorem . For any tree T with order n≥,S(T)≤S(S n–
,n–
).The equality holds
if and only if T∼=Sn– ,n–
.
Claim . For any tree T with order n≥and diameter d≤,S(T) <S(Sn– ,n–
)
except that T∼=S n–
,n–
.
Proof Ifd(T) = , thenT∼=S
a,bfor some positive integersa,bwitha+b=n– . It is well
known that the Laplacian characteristic polynomial ofS
a,bisμ(μ– )n–fa,b(μ), where
fa,b(μ) =μ– (n+ )μ+ (ab+ n+ )μ–n. ()
Note thatSa,bcontainsS,as a subtree. By Lemma .,μ(Sa,b)≥μ(S,) = . Moreover, we know that for any treeT,α(T)≤, with equality if and only ifTis a star. These imply thatμ(Sa,b),μ(Sa,b), andα(Sa,b) consist of the three roots offa,b(μ). As follows from (),
we have
μ
Sa,b+μ
Sa,b+αSa,b=n+ . ()
By virtue of Lemma ., we haveα(S
a,b) >α(Sn– ,n–
) except that (a,b) = (n– ,n– ). Equivalently,S(Sa,b) <S(Sn–
,n–
) except that (a,b) = (n– ,n– ). Ifd(T) = , thenT∼=Sn. We first give a lower bound ofS(Sn–
,n–
) forn≥:
SSn– ,n–
>n+ .. ()
Indeed, by () it suffices to showα(Sn– ,n–
) < .. Note that forn≥,Sn– ,n–
con-tainsS
,as a subtree. By Lemma .,α(Sn– ,n–
)≤α(S ,) =–
Note thatS(Sn) =n+ forn≥. According to (), we haveS(Sn) <S(Sn– ,n–
) for
n≥. As forn∈ {, }, a straightforward calculation shows that
SS,≈., SS, ≈.. ()
Also we haveS(Sn) <S(Sn– ,n–
).
Claim . For any tree T with order n and diameter d≥,S(T) <S(Sn– ,n–
).
Proof Sinced(T)≥, thenn≥ and there is a path of length inT. By inequality (), it suffices to showS(T)≤n+ .. First suppose that there is a pathvv· · ·vinTsuch that eithermax{dT(v),dT(v)} ≥ ormax{dT(v),dT(v)} ≥. LetT,Tbe the two compo-nents ofT–vv. Clearly, bothTandThave at least two edges.
Ifμ,μofT∪Tattain at the same component, sayT, then by Lemma .,
S(T∪T) =S(T)≤m(T) + ≤m(T∪T) + . ()
Note thatS(vv) =S(S) = . By Lemma .,
S(T)≤S(T∪T) +S(vv)≤m(T∪T) + =m(T) + =n+ . ()
Otherwise, S(T ∪ T) = μ(T) + μ(T). Whether max{dT(v),dT(v)} ≥ or
max{dT(v),dT(v)} ≥, we can observe thatmax{d(T),d(T)} ≥. Sayd(T)≥, then
by Corollary .,μ(T) <n(T) – .. By Lemma .(ii),μ(T)≤n(T). Hence,
S(T)≤S(T∪T) +S(vv) =n(T) +n(T) – . + =n+ ..
Next, we may assume that each pathvv· · ·vof length inT hasdT(v) =dT(v) =
anddT(v) =dT(v) = . This implies thatd(T) = andT∼=Sa,bfor some integersa,bwith
a+b=n– . Ifa=b= , thenTis isomorphic to a path of order and a straightforward calculation shows thatS(T) = +√ <n+ ., as claimed. Otherwise, assume without loss of generality thata≥. ThendT(v)≥. LetT,Tbe the two components ofT–vv withvv∈E(T). Then bothTandThave at least two edges. Ifμ,μofT∪Tattain at the same component, sayT, then by Lemmas . and .,
S(T)≤S(T∪T) +S(vv) =S(T) + ≤m(T) + ≤m(T) + =n+ .
Otherwise,S(T∪T) =μ(T) +μ(T). Note thatμ(T)≤n(T). Sinced(T) = , by Corollary .,μ(T) <n(T) – .. So
S(T)≤S(T∪T) +S(vv)≤n(T) +n(T) – . + =n+ ..
Claim . For any tree T with order n and diameter,S(T) <S(Sn– ,n–
).
dT(v)≥. LetT,Tbe the two components ofT–vvwithvv∈E(T). Then bothT andThave at least two edges.
Ifμ,μofT∪Tattain at the same component, sayT, then similarly to inequalities () and (), we can observe thatS(T)≤n+ .
Now letS(T∪T) =μ(T) +μ(T). IfdT(v)≥, thend(T)≥ and henceμ(T) <
n(T) – .. So
S(T)≤S(T∪T) +S(vv) <n(T) +n(T) – . + =n+ ..
IfdT(v) = , thenT∼=Sa,bfor some positive integersa,bwith ≤a+b=n– . Moreover,
sincedT(v)≥, thena≥. If (a,b)∈ {(, ), (, )}, a straightforward calculations show thatS(S
a,b) <n+ .. Otherwise,Sa,bcontains eitherS,orS,as a subtree. Since μ
S, ≈., μ
S,≈.,
it follows from Lemma . thatμ(Sa,b) > .. SinceSa,b is not a star,μn–(Sa,b) < . On
the other hand, note that the matrix ·In– [D(Sa,b) –A(Sa,b)] hasaidentical rows andb
different identical rows, so the multiplicity of eigenvalue is at leasta+b– and else five eigenvalues areμ,μ,μ,μn–andμn= . Since
n
i=μi(Sa,b) = (n– ), we have
i= μi
Sa,b+μn–
Sa,b+μn
Sa,b= (n– ) – (a+b– ) =n+ .
This implies thatS(S
a,b) <n+ –μ(Sa,b) <n+ ..
Next, it suffices to consider the case that each pathvv· · ·vofThasdT(v) =dT(v) = . This implies that T∼=Tk
n for somek≥ andn≥k+ , sinced(T) = . According to
Lemma .,μ(Tnk) =+
√
. Moreover, by Lemma .,
μ
Tnk≤max{di+mi}=n–k– +
n–
n–k– ≤n– +
n– .
Thus forn≥,
STnk≤n– +
n– + +√
<n+ . <S
Sn–
,n–
.
Whenn= ,Tk
n is a path. Comparing with (),S(Tnk) = +
√
<S(S
,). This completes
the proof.
Following from Claims .-., Theorem . holds and the unique tree with maximal
S(T) isSn–
,n– . According to (),
μSn– ,n–
<n+ =mSn– ,n–
+ .
Theorem . Let m,n be two positive integers with n≤m≤n– and Gm,nbe a graph
of order n and size m obtained from a given edge uv by joining m–n+ independent vertices with u and v,respectively,and anothern–m– independent vertices with u.
Proof LetHs,t be a graph obtained by joining a vertex tosvertices of a given complete
graph of order s+tandHc
s,t be its complement graph. ThenHsc,t is isomorphic to the
union ofSt+ andsisolated vertices. Clearly, the Laplacian eigenvalues ofHsc,t consist of
t+ , with multiplicityt– and with multiplicitys+ . Recall that for any graphGwithn
vertices,μi(G) =n–μn–i(Gc) for ≤i≤n– andμn(G) = . So the Laplacian eigenvalues
ofHs,tconsist ofs+t+ with multiplicitys,s+twith multiplicityt– ,sand .
Now Gc
m,nis isomorphic to the union ofHn–m–,m–n+ and an isolated vertex. So the Laplacian eigenvalues ofGc
m,nconsist ofn– with multiplicity n–m– ,n– with
mul-tiplicitym–n, n–m– , and with multiplicity . Therefore, the Laplacian eigenvalues ofGm,nconsist ofn,m–n+ , with multiplicitym–n, with multiplicity n–m–
and . SoS(Gm,n) =n+ (m–n+ ) =m+ .
Recall thatμ(G)≤n(G) for any graphG. Whenm(G) > n(G) – , Haemers’ bound is clearly not attainable. Theorem . implies that ifm(G)≤n(G) – , Haemers’ bound is always sharp for connected graphs other than trees. Ending the paper, we present a conjecture on the uniqueness of the extremal graph.
Conjecture . Among all connected graphs with n vertices and n≤m≤n– edges,
Gm,nis the unique graph with maximal value ofμ+μ.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MG carried out the proofs of the main results in the manuscript. MQZ and YFW participated in the design of the study and drafted the manuscript. All authors read and approved the final manuscript.
Author details
1Department of Mathematics and Physics, Hefei University, Hefei, 230601, China.2School of Mathematical Science,
Chuzhou University, Chuzhou, 239012, China. 3College of Mathematics and Computer Science, Tongling University,
Tongling, 244000, China.
Acknowledgements
The authors are grateful to the referees for carefully reading the manuscript and for providing some comments and suggestions, which led to improvements in the paper. The research was supported by the National Natural Science Foundation of China (11101057, 11201432), the Foundation for Young Talents in College of Anhui Province (2012SQRL170) and the Natural Science Foundation of Anhui Province (1308085MA03).
Received: 23 December 2013 Accepted: 5 June 2014 Published: 23 June 2014 References
1. Bıyıko ˇglu, T, Leydold, J: Algebraic connectivity and degree sequences of trees. Linear Algebra Appl.430, 811-817 (2009)
2. Fallat, S, Kirkland, S: Extremizing algebraic connectivity subject to graph theoretic constraints. Electron. J. Linear Algebra3, 48-74 (1998)
3. Fan, YZ, Xu, J, Wang, Y, Liang, D: The Laplacian spread of a tree. Discrete Math. Theor. Comput. Sci.10(1), 79-86 (2008) 4. Guo, JM: On the second largest Laplacian eigenvalue of trees. Linear Algebra Appl.404, 251-261 (2005)
5. Shao, JY, Zhang, L, Yuan, XY: On the second Laplacian eigenvalues of trees of odd order. Linear Algebra Appl.419, 475-485 (2006)
6. Zhang, XD, Li, JS: The two largest eigenvalues of Laplacian matrices of trees. J. Univ. Sci. Technol. China28, 513-518 (1998)
7. Anderson, WN, Morley, TD: Eigenvalues of the Laplacian of a graph. Linear Multilinear Algebra18, 141-145 (1985) 8. Haemers, WH, Mohammadian, A, Tayfeh-Rezaie, B: On the sum of Laplacian eigenvalues of graphs. Linear Algebra
Appl.432, 2214-2221 (2010)
9. Brouwer, AE, Haemers, WH: Spectra of Graphs. Springer, New York (2012)
10. Fritscher, E, Hoppen, C, Rocha, I, Trevisan, V: On the sum of the Laplacian eigenvalues of a tree. Linear Algebra Appl.
435, 371-399 (2011)
11. Grone, R, Merris, R, Sunder, VS: The Laplacian spectrum of a graph II. SIAM J. Matrix Anal. Appl.11, 218-238 (1990) 12. Merris, R: A note on the Laplacian graph eigenvalues. Linear Algebra Appl.285, 33-35 (1998)
doi:10.1186/1029-242X-2014-242
