Further developments on Brouwer's conjecture for the sum of Laplacian eigenvalues of graphs

Linear Algebra and its Applications 588 (2020) 1–18

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Linear Algebra and its Applications www.elsevier.com/locate/laa

Further developments on Brouwer’s conjecture for the sum of Laplacian eigenvalues of graphs ✩ Hilal A. Ganie a , S. Pirzada a,∗ , Bilal A. Rather a , Vilmar Trevisan b a b

Department of Mathematics, University of Kashmir, Srinagar, Kashmir, India Instituto de Matemática, UFRGS, Brazil

a r t i c l e

i n f o

Article history: Received 13 September 2018 Accepted 21 November 2019 Available online 25 November 2019 Submitted by R. Brualdi MSC: 05C30 05C50 Keywords: Laplacian matrix Laplacian spectrum Clique number Grone-Merris theorem Brouwer’s conjecture c-Cyclic graph

a b s t r a c t Let G be a simple graph with order n and size m and having Laplacian k eigenvalues μ1 , μ2 , . . . , μn−1 , μn = 0 and let Sk (G) = i=1 μi be the sum of k largest Laplacian eigenvalues of G. Brouwer conjectured that Sk (G) ≤ m + k+1 , for all k = 1, 2, . . . , n. We obtain upper bounds for 2 Sk (G), in terms of the clique number ω, the order n and integers p ≥ 0, r ≥ 1, s1 ≥ s2 ≥ 2 associated to the structure of the graph G. We discuss Brouwer’s conjecture for two large families of graphs; the first family of graphs is obtained from a clique of size ω by identifying each of its vertices to a vertex of an arbitrary c-cyclic graph, and the second family is composed of the graphs in which the removal of the edges of the maximal complete bipartite subgraph gives a graph each of whose nontrivial components is a c-cyclic graph. We show among these two large families of graphs, the Brouwer’s conjecture holds for various subfamilies of graphs depending upon the value of c, the order of the c-cyclic graphs, the clique number of the

✩ The research is supported by SERB-DST, New Delhi under the research project number MTR/2017/000084 and CNPq grants 409746/2016-9 and 303334/2016-9, MATHAmSud under project 88881.143281/2017-01 and FAPERGS under project PqG 17/2551-0001. * Corresponding author. E-mail addresses: [email protected] (H.A. Ganie), [email protected] (S. Pirzada), [email protected] (B.A. Rather), [email protected] (V. Trevisan).

https://doi.org/10.1016/j.laa.2019.11.020 0024-3795/© 2019 Published by Elsevier Inc.

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graph, the order of the maximal complete bipartite subgraph and the number of the c-cyclic components of the graph. © 2019 Published by Elsevier Inc.

1. Introduction We consider simple graphs G(V, E) with order n and size m having vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E(G) = {e1 , e2 , . . . , em }. The adjacency matrix A = (aij ) of G is a (0, 1)-square matrix of order n whose (i, j)-entry equal to 1, if vi is adjacent to vj and equal to 0, otherwise. The diagonal matrix associated to G is D(G) = diag(d1 , d2 , . . . , dn ), where di = deg(vi ), for all i = 1, 2, . . . , n. The matrix L(G) = D(G) −A(G) is the Laplacian matrix and its spectrum is the Laplacian spectrum of the graph G. Let 0 = μn ≤ μn−1 ≤ · · · ≤ μ1 be the Laplacian eigenvalues of G. For k = 1, 2, . . . , n, let Sk (G) =

k 

μi ,

(1)

i=1

be the sum of the k largest Laplacian eigenvalues of G. We note that the sum Sk (G) defined by (1) is of much interest by itself and some exciting details, extensions and open problems about it may be found in the excellent paper of Nikiforov [21]. Further, we notice that the investigations of the parameter Sk (G) may turn out to be useful in the study of several fundamental problems in spectral graph theory. In particular, Sk (G) has a strong relation with the extensively studied spectral parameter LE(G) =

n 

|μi −

i=1

2m |, n

defined by Gutman and Zhou [15] as the Laplacian energy of G. A fundamental (and hard) problem is to determine, among all graphs with n vertices, which one has the largest Laplacian energy. Maximum Laplacian energy of unicyclic graphs is discussed in [5]. For more on energy of graphs, we refer to [19]. The parameter Sk (G) is also of great importance in the well known theorem by Grone-Merris [14], a nice proof of which is due to H. Bai [1] and in the Brouwer’s conjecture, which is the subject of this paper. Theorem 1.1. (Grone-Merris-Bai Theorem) If G is any graph of order n and k is any k  positive integer with 1 ≤ k ≤ n, then Sk (G) ≤ d∗i (G), where d∗i (G) = |{v ∈ V (G) : dv ≥ i}|, for i = 1, 2, . . . , n.

i=1

n A strong observation from Theorem 1.1 is that LE(G) ≤ i=1 |d∗i − 2m n |. We note that the equality is attained by threshold graphs and therefore it is worth finding the threshold

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graph having the largest Laplacian energy. Indeed this has been done by Helmberg and Trevisan [17] with the main tool being Theorem 1.1. These observations have increased the evidence to a belief that threshold graphs have large Laplacian energy. Moreover, it supported that the candidate graph for the largest Laplacian energy is a particular pineapple (a graph obtained from a clique on ω vertices by attaching n − ω pendent vertices to a vertex of the clique), an element in this family of graphs. The Brouwer’s conjecture, due to Andries Brouwer [2], is stated as follows. Conjecture 1.2. If G is any graph with order n and size m, then

Sk (G) =

k  i=1

 μi ≤ m +

 k+1 , for any k ∈ {1, 2, . . . , n}. 2

Here we mention about the developments on the Brouwer’s conjecture. Mayank [20] (see also [18]) proved that split graphs and cographs satisfy Brouwer’s conjecture. By using computer computations, Brouwer [2] checked this conjecture for all graphs with at most 10 vertices. For k = 1, the conjecture follows from the well-known inequality μ1 (G) ≤ n and the cases k = n and k = n − 1 are straightforward. Haemers et al. [16] showed that the conjecture is true for all graphs when k = 2 and is also true for trees. Du et al. [6] obtained various upper bounds for Sk (G) and proved that the conjecture is also true for unicyclic and bicyclic graphs. Rocha and Trevisan [24] obtained various upper bounds for Sk (G), which improve the upper bounds obtained in [6] for some cases and proved that the conjecture is true for all k with 1 ≤ k ≤  g5 , where g is the girth of the graph G (the length of the smallest cycle in G). They also showed that the conjecture is true for a connected graph G having maximum degree Δ, p pendant vertices and c cycles with Δ ≥ c + p + 4. Ganie, Alghamdi and Pirzada, [10] obtained upper bounds for Sk (G), in terms of various graph parameters, which improve some previously known upper bounds and showed that the conjecture is true for some new families of graphs [10]. For other progress on Brouwer’s Conjecture, we refer to [3,10–12,18,22] and the references therein. However, Conjecture 1.2 remains open at large. The paper is organized as follows. In Section 2, we give some basic definitions and known results that will be used in the sequel. In Section 3, we obtain upper bounds for Sk (G), in terms of the clique number ω, the order n and integers p ≥ 0, r ≥ 1, s1 ≥ s2 ≥ 2 associated with the structure of the graph G. In Section 4, we discuss the Brouwer’s conjecture for two large families of graphs; the first family of graphs is obtained from a clique of size ω by identifying each of its vertices to a vertex of an arbitrary c-cyclic graph, and the second family is composed of the graphs in which the removal of the edges of the largest complete bipartite subgraph gives a graph each of whose non-trivial components is a c-cyclic graph. We show among these two large families of graphs, the Brouwer’s conjecture holds for various subfamilies of graphs depending upon the value of c, the order of the c-cyclic graphs, the clique number of the graph, the order of the

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maximal complete bipartite subgraph and the number of the c-cyclic components of the graph. 2. Preliminaries In a graph G, a clique is a maximum complete subgraph and the order of the maximum clique is called the clique number of the graph G and is denoted by ω(G) or ω. If H is a subgraph of the graph G, we denote the graph obtained by removing the edges in H from G by G \ H. As usual, Kn and Ks,t denote, respectively, the complete graph on n vertices and the complete bipartite graph on s + t vertices. For other undefined notations and terminology from spectral graph theory, the readers are referred to [4,23]. A very interesting and useful lemma due to Fulton [9] is as follows. Lemma 2.1. Let A and B be two real symmetric matrices both of order n. If k, 1 ≤ k ≤ n, k k k    is a positive integer, then λi (A + B) ≤ λi (A) + λi (B), where λi (X) is the ith i=1

i=1

i=1

eigenvalue of X. The following upper bound for the sum of the k largest Laplacian eigenvalues of a tree T can be found in [7]. Lemma 2.2. Let T be a tree with n ≥ 2 vertices. If Sk (T ) is the sum of the k largest Laplacian eigenvalues of T , then Sk (T ) ≤ n − 2 + 2k −

2k − 2 ≤ n − 2 + 2k, n

for 1 ≤ k ≤ n. For k = 1, equality occurs when G ∼ = K1,n−1 . Assume that a graph G has a special kind of symmetry so that its associated matrix is written in the form ⎛

X ⎜ βt ⎜ . M =⎜ ⎜ .. ⎝ βt βt

β B .. . C C

··· ··· ··· ··· ···

β C .. . B C

⎞ β C⎟ .. ⎟ ⎟ . ⎟, C⎠

(2)

B

where X ∈ Rt×t , β ∈ Rt×s and B, C ∈ Rs×s , such that n = t +cs, where c is the number of copies of B. Then the spectrum of this matrix can be obtained as the union of the spectrum of smaller matrices using the following technique given in [8]. In the statement of the theorem, σ (k) (Y ) indicates the multi-set formed by k copies of the spectrum of Y , denoted by σ(Y ).

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Lemma 2.3. Let M be a matrix of the form given in (2), with c ≥ 1 copies of the block B. Then (i) σ(B − C) ⊆ σ(M ) with multiplicity c − 1;  (ii) σ(M ) \ σ (c−1) ) is the (B − C) = σ(M  set of the remaining t + s eigenvalues of M , √ cβ X  where M = √ t . cβ B + (c − 1)C The next useful lemma found in [25] is as follows. Lemma 2.4. Let X and Y be Hermitian matrices of order n such that Z = X + Y . Then λt (Z) ≤ λj (X) + λt−j+1 (Y ), n ≥ t ≥ j ≥ 1, λt (Z) ≥ λj (X) + λt−j+n (Y ), n ≥ j ≥ t ≥ 1, where λi (M ) is the ith largest eigenvalue of the matrix M . 3. Upper bounds for Sk (G) In this section, we obtain upper bounds for Sk (G), in terms of the clique number ω, the order n and integers p ≥ 0, r ≥ 1, s1 ≥ s2 ≥ 2 associated to the structure of the graph G for some certain types of graphs. Theorem 3.1. Let G be a connected graph of order n ≥ 4 and size m having clique number ω ≥ 2. If H = G \ Kω is a graph having r non-trivial components C1 , C2 , . . . , Cr , each of which is a c-cyclic graph and p ≥ 0 trivial components, then  Sk (G) ≤

ω(ω − 1) + n − p + 2r(c − 1) + 2k, k(ω + 2) + n − p + 2r(c − 1),

if k ≥ ω − 1, if k ≤ ω − 2.

(3)

Proof. Consider the connected graph G of order n ≥ 4 and size m. Let ω ≥ 2 be the clique number of G. Then Kω is a subgraph of G. If we remove the edges of Kω from G, the Laplacian matrix of G can be decomposed as L(G) = L(Kω ∪ (n − ω)K1 ) + L(H), where H = G \ Kω is the graph obtained from G by removing the edges of Kω . Applying Lemma 2.1 and using the fact that Sk (Kω ∪ (n − ω)K1 ) = Sk (Kω ), for 1 ≤ k ≤ n, we have Sk (G) =

k  i=1

μi (G) ≤

k  i=1

μi (Kω ) +

k 

μi (H) = Sk (Kω ) + Sk (H).

i=1

It is well known that the Laplacian spectrum of Kω is {ω (ω−1) , 0}, so that Sk (Kω ) = kω. For k ≥ ω − 1, it is better to consider Sk (Kω ) ≤ ω(ω − 1) and for k ≤ ω − 2, it is better to consider Sk (Kω ) ≤ kω. Therefore, it follows that

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 Sk (Kω ) ≤

ω(ω − 1), kω,

if k ≥ ω − 1, if k ≤ ω − 2.

(4)

In the hypothesis, it is given that H = G \ Kω is a graph having r ≥ 1 non-trivial components, each of which is a c-cyclic graph and p trivial components. Let C1 , C2 , . . . , Cr be the non-trivial components of H with |Ci | = ni ≥ 2 and p be the number of isolated vertices in H. That is, H = H  ∪pK1 , where H  = C1 ∪C2 ∪· · ·∪Cr . Clearly Sk (H) = Sk (H  ) r  and ni = n − p. Let ki be the number of the Laplacian eigenvalues of Ci that belong i=1

to the first k largest Laplacian eigenvalues of H  , where 0 ≤ ki ≤ k, 1 ≤ i ≤ r, and r  ki = k. Since Ci is a c-cyclic graph on ni vertices and mi = ni + c − 1 edges, it follows i=1

that the Laplacian matrix of Ci can be written as L(Ci ) = L(Ti ) + cL(K2 ∪ (ni − 2c)K1 ), where Ti is the spanning tree of Ci . Therefore, applying Lemma 2.2 to Ci , and using Lemma 2.1, we get Ski (Ci ) ≤ Ski (Ti ) + 2c ≤ ni − 2 + 2ki + 2c.

(5)

Now, applying Lemma 2.2 to the graph H  = C1 ∪ C2 ∪ · · · ∪ Cr and using inequality (5), for all ki , 1 ≤ i ≤ r, it follows that 

Sk (H ) = Sk (C1 ∪ C2 ∪ · · · ∪ Cr ) ≤

r 

Ski (Ci )

i=1

≤

r 

(ni − 2 + 2ki + 2c)

i=1

=

r  i=1

ni +

r r   (2c − 2) + 2ki i=1

i=1

= n − p + 2r(c − 1) + 2k. This shows that Sk (H) = Sk (H  ) ≤ n − p + 2r(c − 1) + 2k.

(6)

The result now follows from inequalities (4) and (6). 2 In particular, taking c = 0, we have the following observation, which was obtained in [13]. Corollary 3.2. Let G be a connected graph of order n ≥ 4 and size m having clique number ω ≥ 2. If H = G \ Kω is a forest having r non-trivial components C1 , C2 , . . . , Cr and p ≥ 0 trivial components, then

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 Sk (G) ≤

ω(ω − 1) + n − p − 2r + 2k, k(ω + 2) + n − p − 2r,

if k ≥ ω − 1, if k ≤ ω − 2.

7

(7)

Now, we obtain an upper bound for Sk (G), in terms of the positive integers s1 , s2 and the order n of the graph G. Theorem 3.3. Let G be a connected graph of order n ≥ 4 and size m. Let Ks1 ,s2 , s1 ≥ s2 ≥ 2, be the maximal complete bipartite subgraph of the graph G. If H = G \ Ks1 ,s2 is a graph having r non-trivial components C1 , C2 , . . . , Cr , each of which is a c-cyclic graph and p ≥ 0 trivial components, then  Sk (G) ≤

2s1 s2 + n − p + 2r(c − 1) + 2k, s2 + ks1 + n − p + 2r(c − 1) + 2k,

if k ≥ s1 + s2 − 1, if k ≤ s1 + s2 − 2.

(8)

Proof. Consider the connected graph G with Ks1 ,s2 , (s1 ≥ s2 ), as its maximal complete bipartite subgraph. If we remove the edges of Ks1 ,s2 from G, the Laplacian matrix of G can be decomposed as L(G) = L(Ks1 ,s2 ∪ (n − s1 − s2 )K1 ) + L(H), where H = G \ Ks1 ,s2 is the graph obtained from G by removing the edges of Ks1 ,s2 . Applying Lemma 2.1 and using the fact that Sk (Ks1 ,s2 ∪ (n − s1 − s2 )K1 ) = Sk (Ks1 ,s2 ), for 1 ≤ k ≤ n, we have Sk (G) =

k 

μi (G) ≤

i=1

k  i=1

μi (Ks1 ,s2 ) +

k 

μi (H) = Sk (Ks1 ,s2 ) + Sk (H).

i=1

Now, proceeding similarly as in Theorem 3.1 and using the fact that the Laplacian (s −1) (s −1) spectrum of Ks1 ,s2 is {s1 + s2 , s1 2 , s2 1 , 0}, the result follows. 2 In particular, if s1 = s2 , we have the following consequence of Theorem 3.3. Corollary 3.4. Let G be a connected graph of order n ≥ 4 and size m and let Ks,s , s ≥ 2, be the maximal complete bipartite subgraph of graph G. If H = G \ Ks,s is a graph having r non-trivial components C1 , C2 , . . . , Cr , each of which is a c-cyclic graph and p ≥ 0 trivial components, then  Sk (G) ≤

2s2 + n − p + 2r(c − 1) + 2k, s + k(s + 2) + n − p + 2r(c − 1),

if k ≥ 2s − 1, if k ≤ 2s − 2.

4. Brouwer’s conjecture for some classes of graphs This section is devoted to verify the truth of Brouwer’s conjecture for new families of graphs. Theorem 4.1. Let G be a connected graph of order n ≥ 4 and size m having clique number ω ≥ 2. If H = G \ Kω is a graph having r non-trivial components C1 , C2 , . . . , Cr , each of

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which is a c-cyclic graph, then Sk (G) ≤ m + where Δ1 = min {ω−2, γ1 }, γ1 =

k(k + 1)

, for all k ∈ [1, Δ1 ] and k ∈ [β1 , n],

2  2ω+3− 16ω+8r(c−1)+9 2

and β1 =

3+



4ω 2 −4ω+8r(c−1)+9 . 2

Proof. Let G be a connected graph of order n and size m having clique number ω ≥ 2. If H = G \ Kω is a graph having r non-trivial components C1 , C2 , . . . , Cr , each of which is a c-cyclic graph and p ≥ 0 trivial components, then m = ω(ω−1) + n − p + r(c − 1). 2 For k ≥ ω − 1, from Theorem 3.1, we have Sk (G) ≤ ω(ω − 1) + n − p + 2r(c − 1) + 2k ≤m+

k(k + 1) ω(ω − 1) k(k + 1) = + n − p + r(c − 1) + , 2 2 2

if k2 − 3k − (ω(ω − 1) + 2r(c − 1)) ≥ 0.

(9)

Now, consider the polynomial f (k) = k2 − 3k − (ω(ω − 1)+ 2r(c − 1)). The roots of this  3+

4ω 2 −4ω+8r(c−1)+9

3−

4ω 2 −4ω+8r(c−1)+9

polynomial are β1 = and β2 = . This shows 2 2 that f (k) ≥ 0, for all k ≥ β1 and for all k ≤ β2 . Since 1 ≤ r ≤ ω, it can be seen that 2 − ω < β2 < 3 − ω. Thus, it follows that (9) holds for all k ≥ β1 . It is easy to see that β1 ≥ ω − 1. This completes the proof in this case. For k ≤ ω − 2, from Theorem 3.1, we have k(k + 1) 2 k(k + 1) ω(ω − 1) + n − p + r(c − 1) + , = 2 2

Sk (G) ≤ (ω + 2)k + n − p + 2r(c − 1) ≤ m +

if k2 − (2ω + 3)k − 2r(c − 1) + ω(ω − 1) ≥ 0.

(10)

Proceeding, similarly as above, it can be seen that (10) holds for all k ≤ γ1 =  2ω+3−

16ω+9+8r(c−1) . 2

Indeed, γ1 ≤ ω − 2, holds for any c ≥ 1. This completes the proof in this case as well. 2 Evidently, if c = 0, then Δ1 = ω − 2 for 2 ≤ ω ≤ 5; and Δ1 = γ1 for ω ≥ 6. Further, if c = 1, then Δ1 = ω − 2 for ω = 2; and Δ1 = γ1 for ω ≥ 3. For c ≥ 2, clearly Δ1 = γ1 . From Theorem 4.1, we have the following consequence. Corollary 4.2. Let G be a connected graph of order n ≥ 4 and size m having clique number ω ≥ 2. Let H = G \ Kω be a graph having r non-trivial components C1 , C2 , . . . , Cr , each of which is a c-cyclic graph.

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(i). If c = 0, that is each Ci is a tree, then Conjecture 1.2 holds for all k ∈ [ω + 1, n] √ 2ω+3− 16ω+9−8r and k ∈ [1, Δ1 ], where Δ1 = min {ω − 2, γ1 } and γ1 = . 2 (ii). If c = 1, that is each Ci is a unicyclic graph, then Conjecture 1.2 holds for all √ k ∈ [ω + 2, n] and k ∈ [1, Δ1 ], where Δ1 = min {ω − 2, γ1 } and γ1 = 2ω+3−2 16ω+9 . (iii). If c = 2, that is each C√i is a bicyclic graph, then Conjecture 1.2 holds for all k ∈ [ω + 3, n] and k ∈ [1, 2ω+3− 16ω+9+8r ]. 2 (iv). If c ≥ 3, that is each Ci is a c-cyclic graph, then Conjecture 1.2 holds for all k ∈ [ω + c, n] and k ∈ [1,

2ω+3−

Proof. (i). If c = 0, then β1 = fact 1 ≤ r ≤ ω, we have β1 =

3+

√

(ii). If c = 1, then β1 = β1 =

16ω+9+8r(c−1) ]. 2

√ 3+ 4ω 2 −4ω−8r+9 2

and γ1 =

3+ 4ω 2 − 4ω − 8r + 9 ≤ 2

√ 3+ 4ω 2 −4ω+9 2

3+

√

and γ1 =

3+ 4ω 2 − 4ω + 9 ≤ 2

√ 2ω+3− 16ω+9−8r . 2

√ 4ω 2 − 4ω + 1 = ω + 1. 2

√ 2ω+3− 16ω+9 . 2

√

Using the

We have

4ω 2 + 4ω + 1 = ω + 2. 2

(iii). Proceeding similarly as in part (i) and (ii), we can prove part (iii). (iv). If c ≥ 3, then using r ≤ ω, we have β1 = =

3+ 3+

 

3+ 4ω 2 − 4ω + 8r(c − 1) + 9 ≤ 2



4ω 2 + 4ω(2c − 3) + 9 2

(2ω + (2c − 3))2 − 4c(c − 3) ≤ ω + c. 2 2

Let us now consider some special classes of graphs satisfying the hypothesis of Theorem 4.1. Let Cω (a, a, . . . , a), a ≥ 1 be the family of connected graphs of order n = ω(a +1) and size m obtained by identifying one of the vertex of a c-cyclic graph C of order a + 1 to each vertex of the clique Kω . For the family of graphs Cω (a, a, . . . , a), we see that Brouwer’s conjecture is true for various subfamilies depending upon the value of c, the order of the c-cyclic graphs and the clique number of the graph. Theorem 4.3. For the graph G ∈ Cω (a, a, . . . , a), a ≥ 1, Brouwer’s conjecture holds for all k, if c = 0. If c = 1, Brouwer’s conjecture holds for all k, k = ω + 1 and holds for k = ω + 1, provided a ≤ ω + 1. If c = 2, Brouwer’s conjecture holds for all k, k ∈ / {ω + 1, ω + 2}, holds for k = ω + 1, provided a ≤ ω + 2 and holds for k = ω + 2, provided a ≤ 2ω + 12 . If c ≥ 3, Brouwer’s conjecture holds for all k, provided  1 a ≤ ω − 2 + (2c − 1)ω. Proof. Consider a connected graph G with order n and size m and let G belong to the family Cω (a, a, . . . , a), a ≥ 1. If a = 1, then G is a split graph and the result follows,

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completing the proof in this case. For a = 1, let C be a c-cyclic graph on a + 1 vertices having one of the vertices fused with a vertex of the clique Kω . Let L(C) be the Laplacian matrix of C and let 0 = μa+1 ⎛ ≤ μa ≤ · · · ≤ μ1 be ⎞ the eigenvalues of L(C). Let Cq×q be −1 0 · · · 0 ⎜ 0 0 · · · 0⎟ ⎜ . .. .⎟ ⎟ the matrix defined as Cq×q = ⎜ . · · · .. ⎟ , where q = a + 1. ⎜ .. ⎝ 0 0 · · · 0⎠ 0 0 · · · 0 a+1 By a suitable labelling of vertices of G, it can be seen that the Laplacian matrix of G can be written as ⎞ ⎛ F Cq×q · · · Cq×q F · · · Cq×q ⎟ ⎜ Cq×q ⎜ . . .. ⎟ ⎟ ⎜ .. L(G) = ⎜ .. ··· . ⎟ , where F = L(C) − (ω − 1)Cq×q . ⎝C Cq×q · · · Cq×q ⎠ q×q Cq×q Cq×q · · · F ω Taking X and β to be a matrix of order zero, B = L(C) − (ω − 1)Cq×q and C = Cq×q in (2) and using Lemma 2.3, we have σ(L(G)) = σ (ω−1) (L(C) − ωCq×q ) ∪ σ(L(C)). The eigenvalues of the matrix −ωCq×q are ω with multiplicity one and 0 with multiplicity a. If μa+1 ≤ μa ≤ · · · ≤ μ1 are the eigenvalues of the matrix L(C) − ωCq×q , we see that the eigenvalues of the matrix L(G) are μi , 1 ≤ i ≤ a + 1, each with multiplicity ω − 1 and μi , 1 ≤ i ≤ a + 1. Clearly the graph G \ Kω has ω non-trivial components, each of which is a c-cyclic graph. If c = 0, then each of the ω components of G \ Kω is a tree on a + 1 vertices. Therefore, by Theorem 3.5 in [13], Brouwer’s the conjecture holds for all k, completing the proof in this case. By Lemma 2.1, we first observe that μ1 ≤ ω + μ1 .

(11)

If c = 1, then each of the ω components of G \ Kω is a unicyclic graph on a + 1 vertices. So, by Corollary 4.2, Brouwer’s conjecture holds for all k ≥ ω + 2. Now, we need to show that Brouwer’s conjecture holds for all k ≤ ω + 1. For 1 ≤ k ≤ ω + 1, from (11), it follows that k(k + 1) 2 k(k + 1) ω(ω − 1) +n+ , = 2 2 k2 − (2μ1 + 2ω − 1)k + ω 2 − ω + 2n ≥ 0.

Sk (G) ≤ kμ1 ≤ k(ω + μ1 ) ≤ m +

if

Now, consider the polynomial f (k) = k2 − (2μ1 + 2ω − 1)k + ω 2 − ω + 2n.

(12)

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The discriminant√ of this polynomial is d = (2μ1 − 1)2 + 8ωμ1 − 8n and its roots are √ d d x = (2μ1 +2ω−1)+ and y = (2μ1 +2ω−1)− . This shows that f (k) ≥ 0, for all k ≤ y and 2 2 f (k) ≥ 0, for all k ≥ x. We have y=

(2μ1 + 2ω − 1) −



(2μ1 − 1)2 + 8ωμ1 − 8n ≥ ω, 2

 which implies that 2μ1 − 1 ≥ (2μ1 − 1)2 + 8ωμ1 − 8n, further it implies that μ1 ≤ ωn , which is always true as μ1 ≤ a + 1 = ωn . This shows that (12) holds for all k, k ≤ ω. For t = 2, j = 1 and i = 2, from Lemma 2.4, it follows that μ2 ≤ μ1 . Therefore, for k = ω + 1, we have Sω+1 (G) = (ω − 1)μ1 + 2μ2 ≤ (ω − 1)(ω + μ1 ) + 2μ1 = ω(ω − 1) + (ω + 1)μ1 ≤m+

ω(ω − 1) (ω 2 + 3ω + 2) (ω + 1)(ω + 2) = +n+ , 2 2 2

n n+2ω+1 which is true if μ1 ≤ n+2ω+1 for a ≤ ω + 1, it follows that Conjecture ω+1 . Since ω ≤ ω+1 holds in this case for a ≤ ω + 1. Therefore, completing the proof in this case. If c = 2, each of the ω components of G \ Kω is a bicyclic graph on a + 1 vertices. So, by Corollary 4.2, Brouwer’s conjecture holds for all k ≥ ω + 3. To complete the proof in this case, we need to show that Brouwer’s conjecture holds for all k ≤ ω + 2. For 1 ≤ k ≤ ω, from (11), it follows that

k(k + 1) 2 k(k + 1) ω(ω + 1) +n+ , = 2 2

Sk (G) ≤ kμ1 ≤ k(ω + μ1 ) ≤ m +

if

k2 − (2μ1 + 2ω − 1)k + ω 2 + ω + 2n ≥ 0.

(13)

Now, consider the polynomial f (k) = k2 − (2μ1 + 2ω − 1)k + ω 2 + ω + 2n. The discriminant of √this polynomial is d = √ (2μ1 − 1)2 + 8ωμ1 − 8ω − 8n and its roots (2μ1 +2ω−1)+ d (2μ1 +2ω−1)− d are x = and y = . This shows that f (k) ≥ 0, for all k ≤ y 2 2 and f (k) ≥ 0, for all k ≥ x. We have y=

(2μ1 + 2ω − 1) −

 (2μ1 − 1)2 + 8ω(μ1 − 1) − 8n ≥ ω, 2

 which implies that 2μ1 − 1 ≥ (2μ1 − 1)2 + 8ω(μ1 − 1) − 8n, further it implies that μ1 ≤ ωn + 1, which is always true as μ1 ≤ a + 1 = ωn . This shows that (12) holds for all k, k ≤ ω.

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For t = 2, j = 1 and i = 2, from Lemma 2.4, it follows that μ2 ≤ μ1 . Therefore, for k = ω + 1, we have Sω+1 (G) = (ω − 1)μ1 + 2μ2 ≤ (ω − 1)(ω + μ1 ) + 2μ1 = ω(ω − 1) + (ω + 1)μ1 ≤m+

ω(ω + 1) (ω 2 + 3ω + 2) (ω + 1)(ω + 2) = +n+ , 2 2 2

n n+3ω+1 which is true if μ1 ≤ n+3ω+1 ω+1 . Since ω ≤ ω+1 , for a ≤ ω + 2, it follows that inequality (12) holds for k = ω + 1, provided that a ≤ ω + 2. Also for k = ω + 2, we have

Sω+2 (G) = (ω − 1)μ1 + 3μ2 ≤ (ω − 1)(ω + μ1 ) + 3μ1 = ω(ω − 1) + (ω + 2)μ1 ≤m+

ω(ω + 1) (ω 2 + 5ω + 6) (ω + 2)(ω + 3) = +n+ , 2 2 2

which is true if μ1 ≤ n+4ω+3 ω+2 . Proceeding similarly as above it can be seen that μ1 ≤ n+4ω+3 holds, provided that a ≤ 2ω + 12 and the proof is complete in this case. ω+2 Now, suppose that c ≥ 3. Then each of the ω components of G \ Kω is a c-cyclic graph on a + 1 vertices. So, by Corollary 4.2, Brouwer’s conjecture holds for all k ≥ ω + c. To complete the proof, we need to show that Brouwer’s conjecture holds for all k ≤ ω +c −1. We consider the cases 1 ≤ k ≤ ω and ω + 1 ≤ k ≤ ω + c − 1 one by one. For 1 ≤ k ≤ ω, we have already seen for the case c = 2 that k(k + 1) ω(ω − 1) +n+ω+ 2 2 k(k + 1) k(k + 1) ω(ω − 1) + n + ω(c − 1) + =m+ , < 2 2 2

Sk (G) ≤ kμ1 ≤

as c ≥ 3. This shows that Conjecture 1.2 holds for 1 ≤ k ≤ ω. So, suppose that ω + 1 ≤ k ≤ ω + c − 1. For ω + 1 ≤ k ≤ ω + c − 1, it follows from (11) that Sk (G) ≤ (ω − 1)μ1 + (k − ω + 1)μ2 ≤ (ω − 1)(ω + μ1 ) + (k − ω + 1)μ1 ≤m+ if

ω(ω − 1) k(k + 1) k(k + 1) = + n + ω(c − 1) + , 2 2 2 k2 − (2μ1 − 1)k + (2c − 1)ω + 2n − ω 2 ≥ 0.

Since ω + 1 ≤ k ≤ ω + c − 1, it follows from (14) that (ω + 1)2 − (2μ1 − 1)(ω + c − 1) + (2c − 1)ω + 2n − ω 2 ≥ 0 ⇒

2μ1 ≤

2n + 2cω + 2ω + c . ω+c−1

Using the fact μ1 ≤ a + 1, it follows that

(14)

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2n + 2cω + 2ω + c ω+c−1 (2ω − 1)c + 2ω + 2 . a≤ 2(c − 1)

2a + 2 ≤ ⇔

Since (2ω−1)c+2ω+2 > ω − 1, it follows that inequality (14) always holds for a ≤ ω − 1. 2(c−1) So, we assume that a ≥ ω. Now, consider the polynomial f (k) = k2 − (2μ1 − 1)k + (2c − 1)ω + 2n − ω 2 ,

ω + 1 ≤ k ≤ ω + c − 1.

Using calculus it is clear that f (k) is decreasing for all k ∈ [ω + 1, μ1 − 12 ] and increasing for all k ∈ [μ1 − 12 , ω + c − 1]. Therefore, if f (μ1 − 12 ) ≥ 0, then inequality (14) always holds. We have f (μ1 − 12 ) ≥ 0 if 1 1 (μ1 − )2 − (2μ1 − 1)(μ1 − ) + (2c − 1)ω + 2n − ω 2 ≥ 0 2 2 1  implies that μ1 ≤ + 2n − ω 2 + (2c − 1)ω. 2 For this μ1 , inequality (14) always holds. Since μ1 ≤ a + 1 =

n ω,

we have

1  n ≤ + 2n − ω 2 + (2c − 1)ω ω 2 which implies that 

2a + 1 ≤ 2

2aω − ω 2 + (2c + 1)ω

that is, 4a2 − (8ω − 4)a + (4ω 2 − (8c + 4)ω + 1) ≤ 0, which further gives 1  1  − (2c − 1)ω ≤ a ≤ ω − + (2c − 1)ω. 2 2  This shows that inequality (14) holds for all a, with a ∈ [ω − 12 − (2c − 1)ω, ω − 12 +  (2c − 1)ω]. Since this inequality already  holds for a ≤ ω − 1, it follows that inequality 1 (14) holds for all a, with a ∈ [1, ω − 2 + (2c − 1)ω]. Thus it follows that Conjecture 1.2 holds for all ω + 1 ≤ k ≤ ω + c − 1, provided a ≤ ω − 12 + (2c − 1)ω. Thus completes the proof in this case as well. 2 ω−

Now we turn our attention to the graphs having Ks,s , s ≥ 2 as the maximal complete bipartite subgraph. In this direction we have the following.

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Theorem 4.4. Let G be a connected graph of order n ≥ 4 and size m and let Ks,s , s ≥ 2 be a maximal complete bipartite subgraph of graph G. If H = G \ Ks,s is a graph having r non-trivial components C1 , C2 , . . ., Cr , each of which is a c-cyclic graph and p ≥ 0 5+ 8r(c−1)+34 trivial components, then for s ≥ , Conjecture 1.2 holds for all k; and for 2   5+ 8r(c−1)+34 1 + r(c − 1) + 2 ≤ s < , Conjecture 1.2 holds for all k ∈ [x1 , n] and for 2 all k ∈ [1, y1 ], where x1 =

2s+3+



20s−4s2 +8r(c−1)+9 2

and y1 =

 2s+3− 20s−4s2 +8r(c−1)+9 . 2

Proof. Let G be a connected graph of order n and size m. If H = G \ Ks,s is a graph having r non-trivial components C1 , C2 , . . . , Cr , each of which is a c-cyclic graph and p ≥ 0 trivial components, then m = s2 +n−p+(c−1)r. For 1 ≤ k ≤ n, from Corollary 3.4, we have Sk (G) ≤ s + k(s + 2) + n − p + 2r(c − 1) ≤m+

k(k + 1) k(k + 1) = s2 + n − p + r(c − 1) + , 2 2

if k2 − (2s + 3)k − (2s + 2r(c − 1) − 2s2 ) ≥ 0.

(15)

Now, consider the polynomial f (k) = k2 − (2s + 3)k − (2s + 2r(c − 1) − 2s2 ). The discriminant of this polynomial is d = 20s − 4s2 + 8r(c − 1) + 9. We have d ≤ 0 

if 20s − 4s2 + 8r(c − 1) + 9 ≤ 0, which gives s ≥ 5+



5+

8r(c−1)+34 . 2

This shows that

8r(c−1)+34 for s ≥ , inequality (15) and so Conjecture 1.2 always holds. For s < 2   5+ 8r(c−1)+34 2s+3+ 20s−4s2 +8r(c−1)+9 , the roots of the polynomial f (k) are x1 = and 2 2  2s+3− 20s−4s2 +8r(c−1)+9 y1 = , giving f (k) ≥ 0 for all k ≥ x1 and f (k) ≥ 0 for all k ≤ y1 . 2  5+ 8r(c−1)+34 This shows that for s < , inequality (15) and so Conjecture 1.2 holds for 2

all k ∈ [x1 , n] and for all k ∈ [1, y1 ].

2

The following observations explore some of the concrete families given by Theorem 4.4 for which the Conjecture 1.2 holds. We will use the fact that 1.2 is always true for all k ≤ 2. Corollary 4.5. Let G be a connected graph of order n ≥ 4 and size m and let Ks,s , s ≥ 2 be a maximal complete bipartite subgraph of G. Let H = G \ Ks,s , s ≥ 2 be a graph having r non-trivial components C1 , C2 , . . . , Cr , each of which is a c-cyclic graph. (i). If c = 0, that is, each Ci is a tree, then Conjecture 1.2 holds for all k, if s ≥ 5; and holds for all k, k ∈ / [3, 7], if 2 ≤ s ≤ 4.

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(ii). If c = 1, that is, each Ci is a unicyclic graph, then Conjecture 1.2 holds for all k, if s ≥ 6; and holds for all k, k ∈ / [3, 7], if 2 ≤ s ≤ 5. (iii). If c = 2, that is, each Ci is a bicyclic graph, then Conjecture 1.2 holds for all k, if s ≥ 10; and holds for all k, k ∈ / [3, 12], if 2 ≤ s ≤ 9. (iv). If c = 3, that is, each Ci is a tricyclic graph, then Conjecture 1.2 holds for all k, if s ≥ 14; and holds for all k, k ∈ / [3, 17], if 2 ≤ s ≤ 13. (v). If c = 4, that is, each Ci is a tetracyclic graph, then Conjecture 1.2 holds for all k, if s ≥ 18; and holds for all k, k ∈ / [3, 20], if 2 ≤ s ≤ 17. Proof. (i). If c = 0, then each of the r non-trivial components of H are trees and so Conjecture 1.2 holds for all k if s ≥ 5 and holds for all k, k = 2, 3, 4, 5, 6, 7 if 2 ≤ s ≤ 4 see Theorem 3.9 in [13]. (ii). If c = 1, then each of the r non-trivial components of H are unicyclic graphs and so the discriminant of the polynomial given by the left hand side of (15) becomes d = 20s − 4s2 + 9. Clearly, for s ≥ 6, the discriminant d < 0√ and therefore for such an s, √ Conjecture 1.2 always holds. For s = 5, we have x1 = 13+2 9 = 8 and y1 = 13−2 9 = 5, √ implying that Conjecture 1.2 holds for all k, k = 6, 7. For s = 4, we have x1 = 11+2 25 = 8 √ and y1 = 11−2 25 = 3, giving that Conjecture 1.2 holds for all k, k = 4, 5, 6, 7. For s = 3, √ √ we have x1 = 9+2 33 = 7.3722 and y1 = 9−2 33 = 1.6277, implying that Conjecture 1.2 √ holds for all k, k = 2, 3, 4, 5, 6, 7. For s = 2, we have x1 = 7+2 33 ≈ 6.3723 and y1 = √ 7− 33 ≈ 0.6277, implying that Conjecture 1.2 holds for all k, k = 2, 3, 4, 5, 6, 7. 2 (iii), (iv), (v). These follow by proceeding similar to the above cases. 2 Also, we have the following observations for a connected graph G of order n ≥ 4, size m and having Ks,s , s ≥ 2 as its maximal complete bipartite subgraph. Let H = G \ Ks,s , s ≥ 2 be a graph having r non-trivial components C1 , C2 , . . . , Cr , each of which is a c-cyclic graph. If c, r ≤ 2s , then Conjecture 1.2 holds for all k, if s = 7, s ≥ 9. If s = 8, then Conjecture 1.2 holds for all k for c ≤ 4, r ≤ 3; holds for all k, k = 9, 10 for c = r = 4. If s = 6, then Conjecture 1.2 holds for all k for c ≤ 2, r ≤ 1; holds for all k, for c = r = 2; holds for all k, k = 7, 8 for c = 2, r = 3; holds for all k, k ∈ / [6, 9] for c = 3, r = 2; and holds for all k, k ∈ / [5, 10], for c = r = 3. If s = 5, then Conjecture 1.2 holds for all k, k = 6, 7 for c ≤ 1, r ≤ 2; holds for all k, k ∈ / [5, 8], for c = 2, r ≤ 2. If s = 4, then Conjecture 1.2 holds for all k, k ∈ / [4, 7] for c ≤ 1, r ≤ 2; holds for all k, k ∈ / [3, 8], for c = 2, r ≤ 2. If s = 3, then Conjecture 1.2 holds for all k, k = 3, 4, 5, 6, 7. If s = 2, then Conjecture 1.2 holds for all k, k ∈ / [3, 6]. The following lemma gives a sequence of odd perfect squares. Lemma 4.6. If bt = bt−1 + 8(t + 1), t ≥ 1 with b0 = 0, then xt = bt + 9 is a perfect square. In fact xt = (2t + 3)2 , for all t ≥ 1. Proof. To prove this, we use induction on t. We have x1 = b1 + 9 = b0 + 16 + 9 = 25 = 52 = (2(1) + 3)2 ; x2 = b2 + 9 = b1 + 8(2) + 9 = 49 = 72 = (2(2) + 3)2 . This shows

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that result is true for t = 1 and t = 2. Assume that the result is true for t = u. By induction hypothesis, we have xu = (2u + 3)2 . For t = u + 1, we have xu+1 = bu+1 + 9 = bu +8(u +2) +9 = bu +9 +8(u +2) = xu +8(u +2) = (2u +3)2 +8(u +2) = (2(u +1) +3)2 . This shows that result is true for t = u + 1. Thus it follows by the induction that the result is true for all t ≥ 1. 2 The following observations explore more families given by Theorem 4.4 for which the Conjecture 1.2 holds. Corollary 4.7. Let G be a connected graph of order n ≥ 4 and size m and having Ks,s , s ≥ 2 as its maximal complete bipartite subgraph. Let H = G \ Ks,s , s ≥ 2 be a graph having r non-trivial components C1 , C2 , . . . , Cr , each of which is a c-cyclic graph. (i). If −8 < 20s − 4s2 + 8r(c − 1) ≤ 0, then Conjecture 1.2 holds for all but k ∈ [s + 1, s + 2]. (ii). If bt−1 < 20s − 4s2 + 8r(c − 1) ≤ bt , then Conjecture 1.2 holds for all but k ∈ [s − t, s + t + 3], where bt = bt−1 + 8(t + 1) with t ≥ 1 and b0 = 1. Proof.(i). If −8 < 20s − 4s2 + 8r(c − 1) = 0, then we have s + 2 < x1 = 2s+3+

20s−4s2 +8r(c−1)+9 2

2s+3−

20s−4s2 +8r(c−1)+9

≤ s + 3 and s ≤ y1 = 2 fore the desired result follows from Theorem 4.4. (ii). This follows from Lemma 4.6 and by using Theorem 4.4.

< s + 1. There-

2

We note that the inequality −9 < 20s − 4s2 + 8r(c − 1) ≤ 0 in the unknowns r, c have various solutions. Some of the solutions are r ≤ 2s , c ≤ s − 6; r ≤ s−5 2 , c ≤ s + 1; s−3 s r ≤ s − 5, c ≤ s+2 ; r ≤ s, c ≤ ; r ≤ , c ≤ 2s − 9. 2 2 4 Corollary 4.8. Let G be a connected graph of order n ≥ 4 and size m and having Ks,s , s ≥ 2 as its maximal complete bipartite subgraph. Let H = G \ Ks,s , s ≥ 2 be a graph having r non-trivial components C1 , C2 , . . . , Cr , each of which is a c-cyclic graph. (i). If r ≤ s, then Conjecture 1.2 holds for all but k ∈ [s − c, s + c + 3]. (ii). If c ≤ s, then Conjecture 1.2 holds for all but k ∈ [s − r, s + r + 3]. Proof. If r ≤ s, then d = 20s − 4s2 + 8r(c − 1) + 9 ≤ 12s − 4s2 + 8sc + 9 = 9 + (2c + 3)2 − (2s − 2c − 3)2 < (2c + 4)2 . √

√

d d Therefore, we have x1 = 2s+3+ < 2s+3+2c+4 = s + c + 72 and y1 = 2s+3− > 2 2 2 2s+3−2c−4 1 = s −c − . This shows that Conjecture 1.2 holds for all k, k ∈ / [s −c, s +c +3]. 2 2 Similarly we can prove the other part. 2

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5. Concluding remarks If G is a connected graph which belongs to any of the families of graphs satisfying the hypothesis of Theorem 4.1, then Brouwer’s conjecture holds for all except few values of k. Verification of Brouwer’s conjecture for these few values of k seems to be more difficult, since it depends on the structure of the graph G. It has been proven in [18] that the disjoint union of graphs which satisfy Brouwer’s conjecture also satisfy Brouwer’s conjecture, therefore any disjoint union of graphs that satisfy the hypothesis of theorems given in section 4 also satisfy Brouwer’s conjecture. Recently Helmberg and Trevisan [18] defined spectral threshold dominance property of a graph as follows. Definition 1. A graph G on n vertices with m edges is spectrally threshold dominated if for each k ∈ {1, . . . , n} there is a threshold graph Tk having the same number of vertices k k k and edges satisfying i=1 d∗i (Tk ) = i=1 λi (Tk ) ≥ i=1 λi (G). The authors established its relation with the Brouwer’s conjecture and the Laplacian energy conjecture [18]. More precisely, in the first place, they proved that Brouwer’s conjecture is equivalent to spectrally threshold dominance. At this point, it is an interesting problem to prove directly that a class of graphs is spectrally threshold dominated, which in turns proves that it satisfies Brouwer’s conjecture. This is important because it would increase the number of graphs satisfying Conjecture 1.2, but also, and mainly, because new mathematical tools should emerge from the development. As for the Laplacian energy conjecture, we notice that it has been conjectured that among all connected graphs on n vertices the threshold graph called pineapple with trace  2n 3  maximizes the Laplacian energy (see [26]). Among connected threshold graphs the pineapple is indeed the maximizer; for general threshold graphs on n vertices the clique n−3 of size  2n+1 3  + 1 together with  3  isolated vertices is a threshold graph maximizing Laplacian energy [17] and the conjecture is that this graph has maximum Laplacian energy among all graphs on n vertices. Now, we observe that if Brouwer’s conjecture is true, and hence every graph is spectrally threshold dominated, then the Laplacian energy of every graph is bounded by the energy of threshold graphs, which implies the Laplacian energy conjecture. Declaration of competing interest There is no competing interest. Acknowledgements The authors are highly grateful to the anonymous referee for his valuable comments and suggestions which certainly improved the presentation of the paper. The research

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of S. Pirzada is supported by SERB-DST, New Delhi under the research project number MTR/2017/000084. Vilmar Trevisan acknowledges partial support of CNPq grants 409746/2016-9 and 303334/2016-9, MATHAmSud under project 88881.143281/2017-01 and FAPERGS under project PqG 17/2551-0001. References [1] H. Bai, The Grone-Merris conjecture, Trans. Amer. Math. Soc. 363 (2011) 4463–4474. [2] A.E. Brouwer, W.H. Haemers, Spectra of graphs, Available from http://homepages.cwi.nl/aeb/ math/ipm.pdf. [3] X. Chen, Improved results on Brouwer’s conjecture for sum of the Laplacian eigenvalues of a graph, Linear Algebra Appl. 557 (2018) 327–338. [4] D. Cvetkovic, M. Doob, H. Sachs, Spectra of Graphs-Theory and Application, Academic Press, New York, 1980. [5] K.C. Das, E. Fritscher, L.K. Pinheiro, Vilmar Trevisan, Maximum Laplacian energy of unicyclic graphs, Discrete Appl. Math. 218 (2017) 71–81. [6] Z. Du, B. Zhou, Upper bounds for the sum of Laplacian eigenvalues of graphs, Linear Algebra Appl. 436 (2012) 3672–3683. [7] E. Fritscher, C. Hoppen, I. Rocha, V. Trevisan, On the sum of the Laplacian eigenvalues of a tree, Linear Algebra Appl. 435 (2011) 371–399. [8] E. Fritscher, V. Trevisan, Exploring symmetries to decompose matrices and graphs preserving the spectrum, SIAM J. Matrix Anal. Appl. 37 (1) (2016) 260–289. [9] W. Fulton, Eigenvalues, invariant factors, highest weights and Schubert calculus, Bull. Amer. Math. Soc. (N.S.) 37 (2000) 209–249. [10] H.A. Ganie, A.M. Alghamdi, S. Pirzada, On the sum of the Laplacian eigenvalues of a graph and Brouwer’s conjecture, Linear Algebra Appl. 501 (2016) 376–389. [11] H.A. Ganie, S. Pirzada, Corrigendum to “On the sum of the Laplacian eigenvalues of a graph and Brouwer’s conjecture”, Linear Algebra Appl. 501 (2016) 376–389, Linear Algebra Appl. 538 (1) (2018) 228–230. [12] H.A. Ganie, S. Pirzada, Rezwan Ul Shaban, X. Li, Upper bounds for the sum of Laplacian eigenvalues of a graph and Brouwer’s conjecture, Discrete Math. Algorithms Appl. 11 (2) (2019) 105008, (15 pages). [13] H.A. Ganie, S. Pirzada, Vilmar Trevisan, Brouwer’s conjecture for two families of graphs, preprint. [14] R. Grone, R. Merris, The Laplacian spectrum of a graph II, SIAM J. Discrete Math. 7 (1994) 221–229. [15] I. Gutman, B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414 (2006) 29–37. [16] W.H. Haemers, A. Mohammadian, B. Tayfeh-Rezaie, On the sum of Laplacian eigenvalues of graphs, Linear Algebra Appl. 432 (2010) 2214–2221. [17] C. Helmberg, V. Trevisan, Threshold graphs of maximal Laplacian energy, Discrete Math. 338 (2015) 1075–1084. [18] C. Helmberg, V. Trevisan, Spectral threshold dominance, Brouwer’s conjecture and maximality of Laplacian energy, Linear Algebra Appl. 512 (2017) 18–31. [19] X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012. [20] Mayank, On Variants of the Grone-Merris Conjecture, Master’s thesis, Mathematics and Computer Science, Eindhoven University of Technology, Department of Eindhoven, The Netherlands, November 2010. [21] V. Nikiforov, Extrema of graph eigenvalues, Linear Algebra Appl. 482 (2015) 158–190. [22] S. Pirzada, Hilal A. Ganie, On the Laplacian eigenvalues of a graph and Laplacian energy, Linear Algebra Appl. 486 (2015) 454–468. [23] S. Pirzada, An Introduction to Graph Theory, Universities Press, Orient BlacksSwan, Hyderabad, 2012. [24] I. Rocha, V. Trevisan, Bounding the sum of the largest Laplacian eigenvalues of graphs, Discrete Appl. Math. 170 (2014) 95–103. [25] W. So, Commutativity and spectra of Hermitian matrices, Linear Algebra Appl. 212/213 (1994) 121–129. [26] C. Vinagre, R. Del-Vecchio, D. Justo, V. Trevisan, Maximum Laplacian energy among threshold graphs, Linear Algebra Appl. 439 (5) (2013) 1479–1495.