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On the sum of the two largest Laplacian eigenvalues of trees

Abstract

For S(T), the sum of the two largest Laplacian eigenvalues of a tree T, an upper bound is obtained. Moreover, among all trees with n4 vertices, the unique tree which attains the maximal value of S(T) is determined.

MSC:05C50.

1 Introduction

Let V(G) be the vertex set and E(G) be the edge set of a graph G. The numbers of vertices and edges of G are denoted by n(G) and m(G), respectively. For a vertex vV(G), let N G (v) be the set of vertices adjacent to v and d G (v)=| N G (v)| be the degree of v. Particularly, denote by Δ(G) the maximum degree of G. The diameter of a connected graph G, denoted by d(G), is the maximum distance among all pairs of vertices in G. Let A(G) be the adjacency matrix of G and D(G) be the diagonal matrix of vertex degrees. The matrix D(G)A(G) is called the Laplacian matrix of G and its eigenvalues are called the Laplacian eigenvalues of G. Let μ 1 μ 2 μ n be the Laplacian eigenvalues of a graph G with n vertices. It is well known that μ n =0 and i = 1 n 1 μ i =2m(G). In particular, μ n 1 is called the algebraic connectivity of G and it is denoted by α(G).

The Laplacian matrix is an important topic in the theory of graph spectra. Particularly, much literature has paid attention to μ 1 , μ 2 , μ n 1 or μ 1 μ n 1 for trees (see, for example, [16]). Let S n be the star of order n, S a , b k be the tree obtained from two stars S a + 1 , S b + 1 by joining a path of length k between their central vertices (see Figure 1). As is well known, among all trees of order n, S n has the largest value of μ 1 (see [7]) and S n 3 , 1 1 has the second largest value of μ 1 (see [6]). On the other hand, Guo [4] proved that these two trees also attain the first two smallest values of μ 2 , respectively. This implies that μ 1 , μ 2 cannot 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 of μ 1 + μ 2 . Moreover, Zhang [6] showed that the S k 1 , k 1 1 , S k 1 , k 2 2 , S k 2 , k 2 3 attain simultaneously the largest value of μ 2 among all trees with 2k vertices. Then Shao et al.[5] showed that S k 1 , k 1 2 attains the largest value of μ 2 among all trees with 2k+1 vertices.

Figure 1
figure 1

S a , b k : a tree of order a+b+k+1 .

Another motivation to study the value of μ 1 + μ 2 came from a result of Haemers et al.[8], who showed that μ 1 + μ 2 m(G)+3 for any graph G. This result implies that Brouwer’s conjecture [9],

μ 1 + μ 2 ++ μ k m(G)+ ( k + 1 2 ) ,

is true for k=2. Considering a tree T, we have μ 1 + μ 2 n(T)+2. Recently, Fritscher et al.[10] improved this bound by giving μ 1 + μ 2 <n(T)+2 2 n ( T ) . This paper determines the extremal tree that attains the bound of μ 1 + μ 2 . Moreover, for general connected graphs, we also give a conjecture on the extremal graphs for μ 1 + μ 2 .

2 A sharp upper bound of μ 1 + μ 2

Let S k (G) be the sum of the largest k Laplacian eigenvalues of a graph G. When k=2, we shall write S(G) instead of S k (G) for simplicity. For graphs G and H, we denote by GH the graph with vertex set V(G)V(H) and edge set E(G)E(H). The following lemmas come from an important result as regards a real symmetric matrix.

Lemma 2.1 ([8])

Let G 1 , G 2 ,, G r be some edge-disjoint graphs. Then S k ( i = 1 r G i ) i = 1 r S k ( G i )for any k.

Lemma 2.2 ([8])

For any graph G, S(G)m(G)+3.

Lemma 2.3 Let G be a connected graph, d i = d G ( v i )and m i = Σ v j N G ( v i ) d j / d i . Then

  1. (i)

    [11] μ 1 (G)Δ(G)+1, with equality if and only ifΔ(G)=n(G)1.

  2. (ii)

    [7] μ 1 (G)n(G), with equality if and only if the complement of G is disconnected.

  3. (iii)

    [12] μ 1 (G)max{ d i + m i | v i V(G)}.

Lemma 2.4 ([6])

Let T be a tree of order n. IfT S n , then μ 1 (T) μ 1 ( S n 3 , 1 1 ), with equality if and only ifT S n 3 , 1 1 .

Corollary 2.5 Let T be a tree with n vertices and diameterd3. Then μ 1 (T)<n0.5.

Proof Note that any tree T has diameter d3 if T S n . According to Lemma 2.4, μ 1 (T) μ 1 ( S n 3 , 1 1 ). Further, by Lemma 2.3,

μ 1 ( S n 3 , 1 1 ) max{ d i + m i }=n2+ n 1 n 2 =n1+ 1 n 2 <n0.5

for n5. For n=4, a straightforward calculation shows that μ 1 ( S 1 , 1 1 )=2+ 2 <3.5. □

Lemma 2.6 ([2])

Let T be a tree of order n and diameterd3. Thenα(T)α( S n d + 1 2 , n d + 1 2 d 2 ), with equality if and only ifT S n d + 1 2 , n d + 1 2 d 2 .

Lemma 2.7 ([11])

Let G be a graph with a vertex u of degree one. Thenα(G)α(Gu).

Lemma 2.7 implies that the algebraic connectivity of a tree is not greater than that of its subtree.

Lemma 2.8 ([4])

Let T n k (n2k+1) be a tree obtained from a star S n k by replacing its k edges with k paths of length two, respectively. Ifk2, then μ 2 ( T n k )= 3 + 5 2 .

The following lemma can be found in [13] and is known as the Interlacing Theorem of Laplacian eigenvalues.

Lemma 2.9 Let G be a graph of order n and H be a graph obtained from G by deleting an edge. Then

μ 1 (G) μ 1 (H) μ n (G) μ n (H)=0.

Next we give the main theorem of this section. Its proof is divided into several sequent claims.

Theorem 2.10 For any tree T with ordern4, S(T)S( S n 2 2 , n 2 2 1 ). The equality holds if and only ifT S n 2 2 , n 2 2 1 .

Claim 2.11 For any tree T with ordern4and diameterd3, S(T)<S( S n 2 2 , n 2 2 1 )except thatT S n 2 2 , n 2 2 1 .

Proof If d(T)=3, then T S a , b 1 for some positive integers a, b with a+b=n2. It is well known that the Laplacian characteristic polynomial of S a , b 1 is μ ( μ 1 ) n 4 f a , b (μ), where

f a , b (μ)= μ 3 (n+2) μ 2 +(ab+2n+1)μn.
(1)

Note that S a , b 1 contains S 1 , 1 1 as a subtree. By Lemma 2.9, μ 2 ( S a , b 1 ) μ 2 ( S 1 , 1 1 )=2. Moreover, we know that for any tree T, α(T)1, with equality if and only if T is a star. These imply that μ 1 ( S a , b 1 ), μ 2 ( S a , b 1 ), and α( S a , b 1 ) consist of the three roots of f a , b (μ). As follows from (1), we have

μ 1 ( S a , b 1 ) + μ 2 ( S a , b 1 ) +α ( S a , b 1 ) =n+2.
(2)

By virtue of Lemma 2.6, we have α( S a , b 1 )>α( S n 2 2 , n 2 2 1 ) except that (a,b)=( n 2 2 , n 2 2 ). Equivalently, S( S a , b 1 )<S( S n 2 2 , n 2 2 1 ) except that (a,b)=( n 2 2 , n 2 2 ).

If d(T)=2, then T S n . We first give a lower bound of S( S n 2 2 , n 2 2 1 ) for n6:

S ( S n 2 2 , n 2 2 1 ) >n+1.5.
(3)

Indeed, by (2) it suffices to show α( S n 2 2 , n 2 2 1 )<0.5. Note that for n6, S n 2 2 , n 2 2 1 contains S 2 , 2 1 as a subtree. By Lemma 2.7, α( S n 2 2 , n 2 2 1 )α( S 2 , 2 1 )= 5 17 2 <0.5.

Note that S( S n )=n+1 for n3. According to (3), we have S( S n )<S( S n 2 2 , n 2 2 1 ) for n6. As for n{4,5}, a straightforward calculation shows that

S ( S 1 , 1 1 ) 5.4142,S ( S 2 , 1 1 ) 6.4811.
(4)

Also we have S( S n )<S( S n 2 2 , n 2 2 1 ). □

Claim 2.12 For any tree T with order n and diameterd5, S(T)<S( S n 2 2 , n 2 2 1 ).

Proof Since d(T)5, then n6 and there is a path of length 5 in T. By inequality (3), it suffices to show S(T)n+1.5. First suppose that there is a path v 0 v 1 v 5 in T such that either max{ d T ( v 0 ), d T ( v 5 )}2 or max{ d T ( v 2 ), d T ( v 3 )}3. Let T 1 , T 2 be the two components of T v 2 v 3 . Clearly, both T 1 and T 2 have at least two edges.

If μ 1 , μ 2 of T 1 T 2 attain at the same component, say T 1 , then by Lemma 2.2,

S( T 1 T 2 )=S( T 1 )m( T 1 )+3m( T 1 T 2 )+1.
(5)

Note that S( v 2 v 3 )=S( S 2 )=2. By Lemma 2.1,

S(T)S( T 1 T 2 )+S( v 2 v 3 )m( T 1 T 2 )+3=m(T)+2=n+1.
(6)

Otherwise, S( T 1 T 2 )= μ 1 ( T 1 )+ μ 1 ( T 2 ). Whether max{ d T ( v 0 ), d T ( v 5 )}2 or max{ d T ( v 2 ), d T ( v 3 )}3, we can observe that max{d( T 1 ),d( T 2 )}3. Say d( T 2 )3, then by Corollary 2.5, μ 1 ( T 2 )<n( T 2 )0.5. By Lemma 2.3(ii), μ 1 ( T 1 )n( T 1 ). Hence,

S(T)S( T 1 T 2 )+S( v 2 v 3 )=n( T 1 )+n( T 2 )0.5+2=n+1.5.

Next, we may assume that each path v 0 v 1 v 5 of length 5 in T has d T ( v 0 )= d T ( v 5 )=1 and d T ( v 2 )= d T ( v 3 )=2. This implies that d(T)=5 and T S a , b 3 for some integers a, b with a+b=n4. If a=b=1, then T is isomorphic to a path of order 6 and a straightforward calculation shows that S(T)=5+ 3 <n+1.5, as claimed. Otherwise, assume without loss of generality that a2. Then d T ( v 1 )3. Let T 3 , T 4 be the two components of T v 1 v 2 with v 0 v 1 E( T 3 ). Then both T 3 and T 4 have at least two edges. If μ 1 , μ 2 of T 3 T 4 attain at the same component, say T 3 , then by Lemmas 2.1 and 2.2,

S(T)S( T 3 T 4 )+S( v 1 v 2 )=S( T 3 )+2m( T 3 )+5m(T)+2=n+1.

Otherwise, S( T 3 T 4 )= μ 1 ( T 3 )+ μ 1 ( T 4 ). Note that μ 1 ( T 3 )n( T 3 ). Since d( T 4 )=3, by Corollary 2.5, μ 1 ( T 4 )<n( T 4 )0.5. So

S(T)S( T 3 T 4 )+S( v 1 v 2 )n( T 3 )+n( T 4 )0.5+2=n+1.5.

 □

Claim 2.13 For any tree T with order n and diameter 4, S(T)<S( S n 2 2 , n 2 2 1 ).

Proof First suppose that T contains a path v 0 v 1 v 4 such that max{ d T ( v 1 ), d T ( v 3 )}3. Now n6 and it suffices to show S(T)n+1.5. Without loss of generality assume that d T ( v 1 )3. Let T 1 , T 2 be the two components of T v 1 v 2 with v 0 v 1 E( T 1 ). Then both T 1 and T 2 have at least two edges.

If μ 1 , μ 2 of T 1 T 2 attain at the same component, say T 1 , then similarly to inequalities (5) and (6), we can observe that S(T)n+1.

Now let S( T 1 T 2 )= μ 1 ( T 1 )+ μ 1 ( T 2 ). If d T ( v 2 )3, then d( T 2 )3 and hence μ 1 ( T 2 )<n( T 2 )0.5. So

S(T)S( T 1 T 2 )+S( v 1 v 2 )<n( T 1 )+n( T 2 )0.5+2=n+1.5.

If d T ( v 2 )=2, then T S a , b 2 for some positive integers a, b with 3a+b=n3. Moreover, since d T ( v 1 )3, then a2. If (a,b){(2,1),(3,1)}, a straightforward calculations show that S( S a , b 2 )<n+1.5. Otherwise, S a , b 2 contains either S 4 , 1 2 or S 2 , 2 2 as a subtree. Since

μ 3 ( S 4 , 1 2 ) 1.5068, μ 3 ( S 2 , 2 2 ) 1.5858,

it follows from Lemma 2.9 that μ 3 ( S a , b 2 )>1.5. Since S a , b 2 is not a star, μ n 1 ( S a , b 2 )<1. On the other hand, note that the matrix 1 I n [D( S a , b 2 )A( S a , b 2 )] has a identical rows and b different identical rows, so the multiplicity of eigenvalue 1 is at least a+b2 and else five eigenvalues are μ 1 , μ 2 , μ 3 , μ n 1 and μ n =0. Since i = 1 n μ i ( S a , b 2 )=2(n1), we have

i = 1 3 μ i ( S a , b 2 ) + μ n 1 ( S a , b 2 ) + μ n ( S a , b 2 ) =2(n1)(a+b2)=n+3.

This implies that S( S a , b 2 )<n+3 μ 3 ( S a , b 2 )<n+1.5.

Next, it suffices to consider the case that each path v 0 v 1 v 4 of T has d T ( v 1 )= d T ( v 3 )=2. This implies that T T n k for some k2 and n2k+1, since d(T)=4. According to Lemma 2.8, μ 2 ( T n k )= 3 + 5 2 . Moreover, by Lemma 2.3,

μ 1 ( T n k ) max{ d i + m i }=nk1+ n 1 n k 1 n2+ 2 n 3 .

Thus for n6,

S ( T n k ) n2+ 2 n 3 + 3 + 5 2 <n+1.5<S ( S n 2 2 , n 2 2 1 ) .

When n=5, T n k is a path. Comparing with (4), S( T n k )=4+ 5 <S( S 2 , 1 1 ). This completes the proof. □

Following from Claims 2.11-2.13, Theorem 2.10 holds and the unique tree with maximal S(T) is S n 2 2 , n 2 2 1 . According to (2),

μ ( S n 2 2 , n 2 2 1 ) <n+2=m ( S n 2 2 , n 2 2 1 ) +3.

Theorem 2.14 Let m, n be two positive integers withnm2n3and G m , n be a graph of order n and size m obtained from a given edge uv by joiningmn+1independent vertices with u and v, respectively, and another2nm3independent vertices with u. ThenS( G m , n )=m+3.

Proof Let H s , t be a graph obtained by joining a vertex to s vertices of a given complete graph of order s+t and H s , t c be its complement graph. Then H s , t c is isomorphic to the union of S t + 1 and s isolated vertices. Clearly, the Laplacian eigenvalues of H s , t c consist of t+1, 1 with multiplicity t1 and 0 with multiplicity s+1. Recall that for any graph G with n vertices, μ i (G)=n μ n i ( G c ) for 1in1 and μ n (G)=0. So the Laplacian eigenvalues of H s , t consist of s+t+1 with multiplicity s, s+t with multiplicity t1, s and 0.

Now G m , n c is isomorphic to the union of H 2 n m 3 , m n + 1 and an isolated vertex. So the Laplacian eigenvalues of G m , n c consist of n1 with multiplicity 2nm3, n2 with multiplicity mn, 2nm3, and 0 with multiplicity 2. Therefore, the Laplacian eigenvalues of G m , n consist of n, mn+3, 2 with multiplicity mn, 1 with multiplicity 2nm3 and 0. So S( G m , n )=n+(mn+3)=m+3. □

Recall that μ 1 (G)n(G) for any graph G. When m(G)>2n(G)3, Haemers’ bound is clearly not attainable. Theorem 2.14 implies that if m(G)2n(G)3, 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 2.15 Among all connected graphs with n vertices andnm2n3edges, G m , n is the unique graph with maximal value of μ 1 + μ 2 .

References

  1. Bıyıkoǧlu T, Leydold J: Algebraic connectivity and degree sequences of trees. Linear Algebra Appl. 2009, 430: 811–817. 10.1016/j.laa.2008.09.030

    MathSciNet  Article  Google Scholar 

  2. Fallat S, Kirkland S: Extremizing algebraic connectivity subject to graph theoretic constraints. Electron. J. Linear Algebra 1998, 3: 48–74.

    MathSciNet  Article  Google Scholar 

  3. Fan YZ, Xu J, Wang Y, Liang D: The Laplacian spread of a tree. Discrete Math. Theor. Comput. Sci. 2008,10(1):79–86.

    MathSciNet  Google Scholar 

  4. Guo JM: On the second largest Laplacian eigenvalue of trees. Linear Algebra Appl. 2005, 404: 251–261.

    MathSciNet  Article  Google Scholar 

  5. Shao JY, Zhang L, Yuan XY: On the second Laplacian eigenvalues of trees of odd order. Linear Algebra Appl. 2006, 419: 475–485. 10.1016/j.laa.2006.05.021

    MathSciNet  Article  Google Scholar 

  6. Zhang XD, Li JS: The two largest eigenvalues of Laplacian matrices of trees. J. Univ. Sci. Technol. China 1998, 28: 513–518.

    Google Scholar 

  7. Anderson WN, Morley TD: Eigenvalues of the Laplacian of a graph. Linear Multilinear Algebra 1985, 18: 141–145. 10.1080/03081088508817681

    MathSciNet  Article  Google Scholar 

  8. Haemers WH, Mohammadian A, Tayfeh-Rezaie B: On the sum of Laplacian eigenvalues of graphs. Linear Algebra Appl. 2010, 432: 2214–2221. 10.1016/j.laa.2009.03.038

    MathSciNet  Article  Google Scholar 

  9. Brouwer AE, Haemers WH: Spectra of Graphs. Springer, New York; 2012.

    Book  Google Scholar 

  10. Fritscher E, Hoppen C, Rocha I, Trevisan V: On the sum of the Laplacian eigenvalues of a tree. Linear Algebra Appl. 2011, 435: 371–399. 10.1016/j.laa.2011.01.036

    MathSciNet  Article  Google Scholar 

  11. Grone R, Merris R, Sunder VS: The Laplacian spectrum of a graph II. SIAM J. Matrix Anal. Appl. 1990, 11: 218–238. 10.1137/0611016

    MathSciNet  Article  Google Scholar 

  12. Merris R: A note on the Laplacian graph eigenvalues. Linear Algebra Appl. 1998, 285: 33–35. 10.1016/S0024-3795(98)10148-9

    MathSciNet  Article  Google Scholar 

  13. Godsil C, Royle G: Algebraic Graph Theory. Springer, New York; 2001.

    Book  Google Scholar 

Download references

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).

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Correspondence to Mingqing Zhai.

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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.

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Guan, M., Zhai, M. & Wu, Y. On the sum of the two largest Laplacian eigenvalues of trees. J Inequal Appl 2014, 242 (2014). https://doi.org/10.1186/1029-242X-2014-242

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Keywords

  • upper bound
  • tree
  • Laplacian matrix