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Applied Mathematics Letters 24 (2011) 918–923

Contents lists available at ScienceDirect

Applied Mathematics Letters

journal homepage: www.elsevier.com/locate/aml

Laplacian energy of diameter 3 trees✩

Vilmar Trevisan a,∗, João B. Carvalho a, Renata R. Del Vecchio b, Cybele T.M. Vinagre b

a UFRGS - Instituto de Matemática, Porto Alegre, Brazilb UFF - Instituto de Matemática, Niterói, Brazil

a r t i c l e i n f o

Article history:Received 7 July 2010Received in revised form 30 December 2010Accepted 31 December 2010

Keywords:Laplacian energyTreeAlgebraic connectivity

a b s t r a c t

Let Tn be the set of all trees of diameter 3 and n vertices. We show that the Laplacianenergy of any tree in Tn is strictly between the Laplacian energy of the path Pn and thestar Sn, partially proving the conjecture that this holds for any tree. We also give a totalorder by the Laplacian energy in Tn. Moreover, we show that this order depends onlyon the algebraic connectivity of the tree: the Laplacian energy increases as the algebraicconnectivity decreases in Tn.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

For a simple graph G with vertices v1, . . . , vn and adjacency matrix A, the spectrum of G is the set of eigenvaluesλ1 ≥ λ2 ≥ · · · ≥ λn of A. The energy of a graph G is defined by the quantity

E(G) =

n−i=1

|λi|. (1)

Similarly, the Laplacian spectrum of G is the set of eigenvalues µ1 ≥ µ2 ≥ · · · ≥ µn of the Laplacian matrix of G, givenby L = D − A, where A is the adjacency matrix and D is the diagonal matrix of vertex degrees. It is well known that if G is aconnected graph then the second smallest eigenvalue µn−1 is positive and it is called the algebraic connectivity of G. Givenany graph G with average degree d, the Laplacian energy of G, defined by Gutman and Zhou [1] is given by

LE(G) =

n−i=1

|µi − d|. (2)

An important open problem in the area of spectral graph theory is to determine which graph G among all graphs with nvertices has the maximum energy E(G). This problem has been settled for a few classes of graphs (see, for example [2]). Inparticular, it is known since 1977, due to Gutman [3], that among all trees with n vertices, the path Pn has maximum energy,whereas the star Sn has minimum energy.

For the Laplacian energy, few results are established, so that the extremal energy graphs are not known even for trees.In [4], Radenković and Gutman studied the correlation between the energy and the Laplacian energy of trees. In that paper,the energy and the Laplacian energy for all trees up to 14 vertices are computed. They found that the energy and the Laplacianenergy of a tree are inversely proportional, and formulated the following:

✩ First author was partially supported by CNPq- Grants 309531/2009-08 and 473815/2010-9.∗ Corresponding author. Fax: +55 51 33086173.

E-mail address: trevisan@mat.ufrgs.br (V. Trevisan).

0893-9659/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.aml.2010.12.050

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V. Trevisan et al. / Applied Mathematics Letters 24 (2011) 918–923 919

Conjecture 1. Let Tn be a tree on n vertices. Then

LE(Pn) ≤ LE(Tn) ≤ LE(Sn).

If this conjecture is true then the extremal energy trees for the Laplacian energy are the opposite of the extremal energy trees,a somewhat unexpected result. Moreover it means that the structural factors that increase the energy of a tree, decrease theLaplacian energy and vice versa, a remarkable fact.

Using the set of trees available in Brendan McKay’s web page http://cs.anu.edu.au/~bdm/data/trees.html, we computedthe spectrum and the Laplacian spectrum of all trees with less than 19 vertices. These spectra were arranged in a data basethat can be accessed in the site1 and several experiments may be performed using that data.

In particular, it is possible to reproduce Radenković and Gutman’s experiment and verify that for all the trees in the database, the path has minimum Laplacian energy and the star has maximum Laplacian energy.

In this paper we prove that Conjecture 1 is true for trees of diameter 3. A tree of diameter 3 can be seen as two starswith an edge linking their centers. Throughout this note we will denote a diameter 3 tree with n vertices by T (a, b),where n = a + b + 2 is the number of vertices and a ≥ b ≥ 1 are the leaves at each end of T (a, b). We prove thatLE(Pn) < LE(T (a, b)) < LE(Sn) for any T (a, b).

We also give a total order in the set of all trees of diameter 3, by showing that the Laplacian energy of T (a, b) is strictlydecreasing as a function of a. Moreover, we note that the order by Laplacian energy in this class of trees is solely dependenton the algebraic connectivity of T (a, b), more precisely, the Laplacian energy of T (a, b) increases as its algebraic connectivitydecreases.

2. Preliminaries

In a graph G, a pendant vertex is a vertex of degree 1. A neighbor is a vertex adjacent to a pendant vertex. The following isa result of Faria [5].

Lemma 1. Let p and q be the number of leaves and neighbors of G, respectively. Then 1 is a Laplacian eigenvalue of G of multi-plicity at least p − q ≥ 0.

The next result is due to Brouwer and Haemers [6].

Lemma 2. Let µ1 ≥ µ2 ≥ · · · ≥ µn be the Laplacian eigenvalues of a connected graph G. Then

µi ≥ di − i + 2, i = 1, . . . , n,

where d1 ≥ d2 ≥ · · · ≥ dn is the degree sequence of the vertices of G.

Let T be a treewith n vertices. In order to compute the characteristic polynomial of trees, we refer to [7,8]where amethodwhich operates directly on the tree was proposed. That algorithm can readily be adapted to compute the characteristicpolynomial of the Laplacian matrix. For the sake of completeness we briefly describe the procedure.

The algorithm works by associating, with each vertex v, a rational function a(v) =rs . Here r and s are members of the

polynomial ring Q[λ]. These are computed bottom–up starting with the leaves which are assigned λ − 1 (the tree can berooted in an arbitrary way). Once all children of v have been processed, v is assigned the function

a(v) = λ − dv −

−c∈C

1a(c)

, (3)

where C is the set of its children and dv is the degree of v. After all vertices have been processed, we compute the charac-teristic polynomial by taking the product of all functions a(v):

p(λ) =

∏v∈V

a(v). (4)

Lemma 3. The Laplacian characteristic polynomial of T (a, b) is

p(λ) = λ (λ − 1)n−4[λ3

+ (−4 − b − a)λ2+ (ab + 2 a + 5 + 2 b)λ − b − a − 2].

Proof. This is an application of the algorithm described above. �

3. Ordering by the Laplacian energy

Let T = T (a, b) be a tree with n = a + b + 2 vertices and diameter 3, with a ≥ b ≥ 1. We notice that for any tree, itsaverage degree d =

2(n−1)n = 2 −

2n .

Lemma 4. T (a, b) has exactly 2 eigenvalues greater than d = 2 −2n .

1 http://www2.mat.ufrgs.br/graphenergy/.

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Proof. By Lemma 1, we notice that the eigenvalue 1 has multiplicity at least a + b − 2 = n − 4. It is also well known thatµn = 0, and that µn−1 (the algebraic connectivity) is smaller than 1. Therefore, there are at least n − 2 eigenvalues smallerthan d.

The fact that there are 2 eigenvalues greater than d follows from Lemma 2, since

µ1 ≥ d1 + 1 = a + 2 ≥ 3,µ2 ≥ d2 = b + 1 ≥ 2.

We remark that this proof can also be obtained by examining the characteristic polynomial p(λ) of T (a, b), given byLemma 3. �

Consider now the n-vertex class of trees T (a + k, b − k), for k = 0, . . . , b − 1.

Theorem 1. The Laplacian energy of T (a + k, b − k) is a strictly decreasing function of k, for k = 0, . . . , b − 1.

Proof. Let T = T (a + k, b − k) and T ′= T (a + ℓ, b − ℓ), with ℓ > k ≥ 0. Let us denote the eigenvalues of T and T ′ by

µ1 ≥ µ2 ≥ · · · ≥ µn and µ′

1 ≥ µ′

2 ≥ · · · ≥ µ′n, respectively. According to Lemmas 4 and 3, we notice that µi = µ′

i , for alli, except for i = 1, 2 and n − 1. Therefore

LE(T ) − LE(T ′) = (d − µn−1) − (d − µ′

n−1) + (µ1 − d) − (µ′

1 − d) + (µ2 − d) − (µ′

2 − d)

= (µ1 + µ2) − (µ′

1 + µ′

2) + (µ′

n−1 − µn−1). (5)

Using Lemma 3, we can compute the characteristic polynomials of T and T ′ and also the degree 3 polynomials q(λ) andq′(λ) whose roots are µ1, µ2, µn−1 and µ′

1, µ′

2, µ′

n−1, respectively. The coefficients of λ2 of q(λ) and q′(λ) are −[(a + k) +

(b − k) + 4] and −[(a + ℓ) + (b − ℓ) + 4], respectively. In any case they equal n + 2. Hence, we have

µ1 + µ2 + µn−1 = n + 2 = µ′

1 + µ′

2 + µ′

n−1,

or

(µ1 + µ2) − (µ′

1 + µ′

2) + (µn−1 − µ′

n−1) = 0. (6)

Now inserting Eq. (6) into Eq. (5) we obtain

LE(T ) − LE(T ′) = 2(µ′

n−1 − µn−1).

Following the result of Grone and Merris [9], it is known that µ′

n−1 < µn−1. Therefore, LE(T ) − LE(T ′) < 0 and the resultfollows. �

Corollary 1. Let n > 3 be an integer. Among all trees T (a, b) of diameter 3 and n vertices, T (n − 2, 1) has minimum Laplacianenergy, whereas T (⌈(n − 2)/2⌉ , ⌊(n − 2)/2⌋) has maximum Laplacian energy.

We observe that the order of the Laplacian energy of the trees T (a, b) depends only on the algebraic connectivity ofT (a, b).

Theorem 2. Let T = T (a, b) and T ′= T (a′, b′) be two trees of diameter 3 and n > 3 vertices with Laplacian spectrum given by

µi, and µ′

i , respectively, for i = 1, . . . , n. The following are equivalent:(i) µn−1 > µ′

n−1;(ii) LE(T ) < LE(T ′);(iii) a > a′.

Proof. From the proof of Theorem 1, we notice that LE(T ) − LE(T ′) = 2(µ′

n−1 − µn−1). This proves the equivalence (i) and(ii). The implication (iii) → (ii) is given by Theorem 1. The implication (ii) → (iii) can be proven by contradiction using thesame argument of Theorem 1. �

The above result says that the ordering of diameter 3 trees by the Laplacian energy solely depends on the algebraicconnectivity. More precisely, the Laplacian energy increases as the algebraic connectivity decreases.

4. Extremal Laplacian energy trees

The next result plays an important role to prove that the star bounds the Laplacian energy among all trees with n verticesand diameter 3.

Lemma 5. The algebraic connectivity µn−1 of T (a, b) satisfies

µn−1 >2n.

Proof. Clearly the algebraic connectivity of T (a, b) is the smallest root of the factor

q(λ) = λ3+ (−4 − b − a)λ2

+ (ab + 2 a + 5 + 2 b)λ − b − a − 2,

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of the characteristic polynomial given by Lemma 3. By Lemma 4 we know that q(x) has two roots greater than d and sinceq(0) = −b−a−2 < 0, it suffices to prove that q(2/n) < 0. Nowby direct computationwe see that q(2/n) = q(2/(a+b+2))is equal to

−−4 ab + 2 b2 + 2 a2 + 2 a2b2 + 4 a2b + 2 ab3 + 4 ab2 + 4 a3 + 4 b3 + b4 + 2 a3b + a4

(a + b + 2)3,

which is clearly negative, since (for example), 4ab2 − 4ab ≥ 0 for b ≥ 1. �

Theorem 3. Let Sn be the star on n vertices. Then

LE(Sn) > LE(T (a, b)).

Proof. It is well known that the Laplacian spectrum of Sn is {0, 1n−2, n}, hence

LE(Sn) = d + (d − 1)(n − 2) + (n − d).

By Lemma 4,

LE(T (a, b)) = d + (d − µn−1) + (d − 1)(n − 4) + (µ2 − d) + (µ1 − d). (7)

Hence,

LE(Sn) − LE(T (a, b)) = (n − d) + 2(d − 1) − (µ1 − d) − (µ2 − d) − (d − µn−1)

= n − 2 + 2d + µn−1 − µ2 − µ1

= n + 2 −4n

+ µn−1 − µ2 − µ1.

Now, since µn−1, µ2 and µ1 are the roots of

q(λ) = λ3+ (−4 − b − a)λ2

+ (ab + 2 a + 5 + 2 b)λ − b − a − 2,

we see that

µn−1 + µ2 + µ1 = a + b + 4 = n + 2,

and hence

LE(Sn) − LE(T (a, b)) = 2µn−1 −4n.

By Lemma 5, µn−1 > 2n , and the result follows. �

We now proceed to prove that the Laplacian energy of the path Pn on n vertices is smaller than the Laplacian energy ofT (a, b).

Lemma 6. Let n > 2. Then

cosπ in

>1n,

for i = 1, . . . , ⌊n/2⌋.

Proof. Since the cosine function is decreasing in the interval being considered, it suffices to show that cos π in > 1

n , for thelargest integral value of i < n/2. If n is even, such value is i = (n − 2)/2, whereas, if n is odd, then i = (n − 1)/2.

Let n be odd. Then cos πn

n−12 = sin π

2n . Now, because sin(x) ≥ x −x33! , we see that

sinπ

2n−

1n

≥n2(24π − 48) − π3

48n3,

which is positive for n > 2. Similarly for n even, cos πn

n−22 = sin π

n and

sinπ

n−

1n

≥n2(6π − 6) − π3

6n3

is positive for n > 2. �

Lemma 7. Let Pn be the path on n vertices. Then

LE(Pn) = 2 + 4⌊n/2⌋−i=1

cosπ in

+1n

(−1)n − 1

.

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Proof. It is well known that the Laplacian spectrum of the path Pn is given by

µi = 2 + 2 cosπ in

, i = 1, . . . , n − 1, µn = 0.

Thus we may write

LE(Pn) = d +

n−1−i=1

d − 2 − 2 cosπ in

= d + 2n−1−i=1

1n + cosπ in

.Using the fact that cos π

n i = − cos πn (n − i) and Lemma 6 we see that

LE(Pn) = d + 4⌊n/2⌋−i=1

cosπ in

+ ϵ,

where ϵ =

2n

if n is even

0 if n is odd. Observing now that d = 2 −

2n , the result follows. �

Theorem 4. Let Pn be the path on n vertices. Then

LE(Pn) < LE(T (a, b)).

Proof. Our experiments (and also those of Radenković and Gutman [4]) show that this result is true for all values of n ≤ 18.So we may assume that n > 18.

Using Eq. (7), we see that

LE(T (a, b)) = d + (d − µn−1) + (d − 1)(n − 4) + (µ2 − d) + (µ1 − d)

= (d − 1)(n − 4) − µn−1 + µ2 + µ1.

Using the fact that µn−1 + µ2 + µ1 = a + b + 4 = n + 2 and noticing that d = 2 −2n , we conclude that

LE(T (a, b)) = 2n +8n

− 4 − 2µn−1. (8)

Now, by using Riemann’s sum to approximate integrals, it is possible to prove that

1 =

∫ π2

0sin x dx ≥

π

n

⌊ n2⌋−

j=1

cosπ jn

.

From this we see that∑⌊ n

2⌋j=1 cos π j

n ≤nπ. Hence, using Lemma 7, we write

LE(Pn) ≤ 2 + 4nπ

.

Thus, using Eq. (8),

LE(Pn) − LE(T (a, b)) ≤ n

4π

− 2

+ 6 + 2µn−1 −8n.

Because µn−1 < 1 and 8n > 0, it follows that

LE(Pn) − LE(T (a, b)) ≤ n

4π

− 2

+ 8,

which is negative since n > 18. �

5. Conclusions

We have shown that Laplacian energy of any tree of diameter 3 and n vertices is strictly between the Laplacian energy ofthe path Pn and the Laplacian energy of the star Sn, a step towards the proof of Conjecture 1. We also gave a total orderingby the Laplacian energy on the set of diameter 3 trees.

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V. Trevisan et al. / Applied Mathematics Letters 24 (2011) 918–923 923

Fig. 1. Laplacian spectrum of trees of seven vertices.

Our data base available at our site2 is a useful tool to study the spectra of trees and to formulate conjectures. As anexample, we show in Fig. 1 the data for the Laplacian spectrum of all trees of seven vertices. The trees can be ordered bytheir Laplacian energy and other invariants, such as diameter. We noticed that at least 4 eigenvalues are smaller than theaverage degree. This seems to be the case for all n and we state the following:

Conjecture 2. For a tree T with n vertices, the number of Laplacian eigenvalues smaller than the average degree d is at least⌈n/2⌉.

Remark. Lemma 4 proves this conjecture for diameter 3 trees. Moreover, if Tk is a tree with diameter D = k + 2, whichis a path Pk with a pendant vertices in an extremity and b in the other, and if k ≤ ⌊n/2⌋, then the conjectures followseasily. In fact, it is sufficient to note that 1 is a Laplacian eigenvalue of Tk with multiplicity at least a + b − 2 (Lemma 1).Considering the algebraic connectivity and 0, there are at least a + b Laplacian eigenvalues smaller than d. Since in Tk wehave n = a + b + k, a + b = n − k > ⌈n/2⌉.

The conjecture is also true if we have a caterpillar with n = k +∑k

i=1 ai vertices and diameter D = k + 2, wherek ≤ ⌊n/4⌋ and the integers satisfy ai ≥ 1, for all 1 ≤ i ≤ k. The multiplicity of 1 as a Laplacian eigenvalue of this graph is atleast

∑ki=1 ai − k = n− 2k, by Lemma 1. Computing 0 and the algebraic connectivity, we have at least n− 2k+ 2 Laplacian

eigenvalues smaller than d and n − 2k + 2 ≥ ⌈n/2⌉ + 2.

References

[1] I. Gutman, B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414 (2006) 29–37.[2] I. Gutman, The energy of a graph: old and new results, in: A. Betten, A. Kohnert, R. Laue, A.Wassermann (Eds.), Algebraic Combinatorics andApplications,

Springer-Verlag, 2001, pp. 196–211.[3] I. Gutman, Acyclic system with extremal Hückel π-electron energy, Theor. Chim. Acta 45 (1977) 79–87.[4] S. Radenković, I. Gutman, Total π-electron energy and Laplacian energy: how far the analogy goes? J. Serb. Chem. Soc. 72 (12) (2007) 1343–1350.[5] I. Faria, Permanental roots and the star degree of a graph, Linear Algebra Appl. 64 (1985) 255–265.[6] A.E. Brouwer, W.H. Haemers, A lower bound for the Laplacian eigenvalues of a graph: proof of a conjecture by Guo, Linear Algebra Appl. 429 (2008)

2131–2135.[7] G.H. Fricke, S.T. Hedetniemi, D.P. Jacobs, V. Trevisan, Reducing the adjacency matrix of a tree, Electron. J. Linear Algebra 1 (1996) 34–43.[8] D.P. Jacobs, V. Trevisan, Constructing the characteristic polynomial of a tree’s adjacency matrix, Congr. Numer. 134 (1998) 139–145.[9] R. Grone, R. Merris, Ordering trees by algebraic connectivity, Graphs Combin. 6 (1990) 229–237.

2 http://www2.mat.ufrgs.br/graphenergy/.