Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs

Ke Wang; Zhen Lin; Shumin Zhang; Chengfu Ye

doi:10.1515/math-2024-0062

Article Open Access

Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs

Ke Wang , Zhen Lin , Shumin Zhang and Chengfu Ye

Published/Copyright: October 22, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

Let G be a graph with n ( G ) vertices and e ( G ) edges, and S k ( G ) be the sum of the k largest Laplacian eigenvalues of G . Brouwer conjectured that S k ( G ) ≤ e ( G ) + k + 1 2 for 1 ≤ k ≤ n ( G ) . In this article, we obtain upper bounds of S k ( G ) in terms of the graphs that contain the friendship graph or the book graph as a subgraph. Further, we show that Brouwer’s conjecture holds for Halin graphs and certain classes of graphs.

Keywords: Brouwer’s conjecture; Laplacian eigenvalues; Halin graph

MSC 2010: 05C50; 05C07; 05C09

1 Introduction

The graphs discussed in this article are finite, undirected and simple. Brouwer’s conjecture states that any graph G = ( V , E ) with Laplacian matrix L and eigenvalues μ 1 ( G ) ≥ μ 2 ( G ) ≥ … ≥ μ n ( G ) = 0 satisfies ∑ i = 1 k μ i ≤ ∣ E ∣ + k + 1 2 . There have been many partial progresses on this conjecture using particular methods from matrix theory. Up to now, the conjecture has been proved to be true for all graphs with at most ten vertices [1], any graphs when k = 2 , n − 2 , n − 3 [2,3], trees [3], threshold graphs [4], unicyclic graphs [5], bicyclic graphs [5,6], tricyclic graphs having no pendent vertices [6,7], regular graphs [8,9], split graphs [9], cographs [8,9], planar graphs when k ≥ 11 , and bipartite graphs when k ≥ 32 n [10]. Li and Guo [11] proposed the full Brouwer’s conjecture and proved that the conjecture holds when k = 2 . Recently, Torres and Trevisan [12] proved that Brouwer’s conjecture holds for the cartesian product of graphs. For more details, we refer to [13–24].

The rest of the article is organized as follows. In the next section, we recall some basic notions and lemmas used further. In Section 3, we obtain upper bounds of S k ( G ) in terms of the friendship subgraph and book subgraph. Moreover, we show that Brouwer’s conjecture holds for it. In Section 4, we show that Brouwer’s conjecture holds for the Halin graph. In Section 5, we show that Brouwer’s conjecture for some classes of graphs.

2 Preliminaries

Let G = ( V , E ) be a graph with the vertex set V ( G ) and edge set E ( G ) , and let ∣ V ( G ) ∣ = n ( G ) and ∣ E ( G ) ∣ = e ( G ) . The Laplacian matrix of G , denoted by L ( G ) , is given by L ( G ) = D ( G ) − A ( G ) , where D ( G ) is the diagonal matrix whose entries are the degrees of the vertex of G and A ( G ) is the adjacency matrix of G . The multiset of the eigenvalues of L ( G ) , { μ 1 ( G ) , μ 2 ( G ) , … , μ n − 1 ( G ) , μ n ( G ) } is called the Laplacian spectrum of G . Without loss of generality, assume that μ 1 ( G ) ≥ μ 2 ( G ) ≥ … ≥ μ n ( G ) = 0 . The sum of the k largest Laplacian eigenvalues of a graph G is defined as S k ( G ) = ∑ i = 1 k μ i ( G ) , where 1 ≤ k ≤ n . The investigation of S k ( G ) is mainly to confirm the following conjecture.

Conjecture 2.1

(Brouwer’s conjecture [1]) For any graph G with n vertices and for any k ∈ { 1 , 2 , … , n } ,

S k ( G ) ≤ e ( G ) + k + 1 2 .

The degree of a vertex v ∈ V ( G ) is the number of vertices in G that are adjacent to v . A pendant vertex is a vertex of degree 1. A pendant edge is an edge incident to a pendant vertex. A graph H is a subgraph of G , H ⊆ G , if V ( H ) ⊆ V ( G ) and E ( H ) ⊆ E ( G ) . In particular, H is a spanning subgraph of G if V ( H ) = V ( G ) . We use the notation i ∼ j to mean the existence of an edge between the vertices i and j of G . The Cartesian product of G 1 and G 2 , denoted by G 1 □ G 2 , is the graph, where ( u i , v j ) ∼ ( u r , v s ) if either u i = u r in G 1 and v j ∼ v s in G 2 or u i ∼ u r in G 1 and v j = v s in G 2 . The join of graphs G 1 and G 2 is the graph G 1 ∨ G 2 obtained from G 1 ∪ G 2 by joining each vertex of G 1 with every vertex of G 2 . For any H ⊆ G , the graph G \ H is the graph obtained from G by removing the edges of H . The p -book graph is defined as the graph cartesian product B p = K 1 , p □ P 2 with p ≥ 3 , where K 1 , p is a star graph and P 2 is the path graph on two notes. In particular, B 2 = t K 1 ∨ P 2 . As usual, K n , K 1 , n − 1 , and P n denote the complete graph, a star graph, and a path of order n , respectively.

A graph with 2 t + 1 vertices consisting of t triangles that intersect in exactly one common vertex is called a t friendship graph, denoted by F t (Figure 1).

$Figure 1 Friendship graph and book graph.$

Figure 1

Friendship graph and book graph.

For any S ⊆ V ( G ) , the subset S is said to be a covering set of G if every edge of G is incident to at least one vertex in S . A covering set with the minimum cardinality among all covering sets is called the minimum covering set of G and its cardinality is called the covering number of G , denoted by τ ( G ) (simply τ ).

Lemma 2.2

[3] Let T n be a tree with n vertices. Then, S k ( T n ) ≤ e ( T n ) + 2 k − 1 for 1 ≤ k ≤ n .

Let the graph have a special kind of symmetry so that its associated matrix is written in the form

(1) M = X β ⋯ β β β t B ⋯ C C ⋮ ⋮ ⋯ ⋮ ⋮ β t C ⋯ B C β t C ⋯ C B ,

where X ∈ R t × t , β ∈ R t × s , and B , C ∈ R s × s , such that n = t + c s , where c is the number of copies of B . Let σ ( k ) ( M ) indicate the multiset formed by k copies of spectrum of M .

Lemma 2.3

[25] Let M be a matrix of the form given in (2.1), with c ≥ 1 copies of block B . Then,

σ ( B − C ) ⊆ σ ( M ) with multiplicity c − 1 ;
σ ( M ) \ σ c − 1 ( B − C ) = σ ( M ′ ) is the set of the remaining t + s eigenvalues of M ′ , where
M ′ = X c β c β t B + ( c − 1 ) C .

Lemma 2.4

[26] Let A and B be two real symmetric matrices of size n. Then, for any 1 ≤ k ≤ n ,

∑ i = 1 k λ i ( A + B ) ≤ ∑ i = 1 k λ i ( A ) + ∑ i = 1 k λ i ( B ) .

3 Upper bounds for S k ( G )

Let Γ 1 be the family of all connected graphs except for the graphs G , where the vertices in the covering set C 1 of the subgraph F t have the property that there are pendent vertices incident to some v ∈ C 1 or any two vertices of C 1 form a triangle with a vertex u ∈ V ( G ) \ C , where C is the covering set of G .

Theorem 3.1

Let G ∈ Γ 1 be a connected graph of n vertices and m edges with covering number τ . If a friendship graph F t is a subgraph of G, then Brouwer’s conjecture holds for G for all k ∈ [ 1 , x 1 ] and k ∈ [ x 2 , n ] when τ ≤ 3 t 2 + 1 , where x 1 = ( 2 τ − 2 t + 3 ) − ( 2 τ − 2 t + 3 ) 2 − 4 ( 4 + 2 t ) 2 , x 2 = ( 2 τ − 2 t + 3 ) + ( 2 τ − 2 t + 3 ) 2 − 4 ( 4 + 2 t ) 2 .

Proof

Let F t be a friendship graph and F t ⊆ G . Clearly, τ ( F t ) = t + 1 . Let C = { v 1 , v 2 , … , v τ } be a minimum covering set of G , and C 1 = { v 1 , v 2 , … , v t + 1 } ⊆ C be a minimum covering set of F t . Since the Laplacian spectrum of F t is { 2 t + 1 , 3 ( t ) , 1 ( t − 1 ) , 0 } ,

(2) S k ( F t ) = 2 t + 3 k − 2 , 1 ≤ k ≤ t + 1 , 4 t + k , t + 2 ≤ k ≤ n − 1 .

By Lemma 2.4, we have S k ( G ) ≤ S k ( F t ) + S k ( G \ F t ) . In order to establish the theorem, we consider the value of S k ( G \ F t ) .

For each i = t + 2 , t + 3 , … , τ , let G i be a spanning subgraph of H = G \ F t with the edge set E ( G i ) = { u v i ∣ u ∈ N ( v i ) \ { v 1 , v 2 , … , v i − 1 } , v i ∈ C \ C 1 } and ∣ E ( G i ) ∣ = m i . Clearly, G i = K 1 , m i ∪ ( n ( H ) − m i − 1 ) K 1 , for i = t + 2 , t + 3 , … , τ . Note that n ( G i ) = n ( H ) , for each i . Since the Laplacian spectrum of G i is { m i + 1 , 1 [ n ( G i ) − 2 ] , 0 [ n ( H ) − m i ] } , S k ( G i ) ≤ m i + k , for i = t + 2 , t + 3 , … , τ .

Note that ∑ i = t + 2 τ m i = m − 3 t , thus

(3) S k ( H ) ≤ ∑ i = t + 2 τ S k ( G i ) ≤ ∑ i = t + 2 τ m i + k = m − 3 t + ( τ − t − 1 ) k .

For 1 ≤ k ≤ t + 1 , we have

S k ( G ) ≤ m − t + ( τ − t + 2 ) k − 2 ≤ m + k 2 + k 2 ,

(4) k 2 − ( 2 τ − 2 t + 3 ) k + 4 + 2 t ≥ 0 .

Consider the polynomial f ( k ) = k 2 − ( 2 τ − 2 t + 3 ) k + 4 + 2 t . The roots of this polynomial are x 1 = ( 2 τ − 2 t + 3 ) − ( 2 τ − 2 t + 3 ) 2 − 4 ( 4 + 2 t ) 2 and x 2 = ( 2 τ − 2 t + 3 ) + ( 2 τ − 2 t + 3 ) 2 − 4 ( 4 + 2 t ) 2 . This shows that f ( k ) ≥ 0 , for all k ≥ x 2 and for all k ≤ x 1 . It is easy to see that x 1 ≥ 1 and x 2 ≤ t + 1 when τ ≤ 3 t 2 + 1 .

For k ≥ t + 2 , we have

S k ( G ) ≤ m + t + ( τ − t ) k ≤ m + k 2 + k 2 ,

(5) k 2 − [ 2 ( τ − t ) − 1 ] k − 2 t ≥ 0 .

Consider the polynomial f ( k ) = k 2 − [ 2 ( τ − t ) − 1 ] k − 2 t . The roots of this polynomial are x 3 = ( 2 τ − 2 t − 1 ) − ( 2 τ − 2 t − 1 ) 2 + 8 t 2 and x 4 = ( 2 τ − 2 t − 1 ) + ( 2 τ − 2 t − 1 ) 2 + 8 t 2 . This shows that f ( k ) ≥ 0 , for all k ≥ x 4 and for all k ≤ x 3 . It is easy to see that x 3 < 0 and x 4 ≤ t + 2 when τ ≤ 3 t 2 + 1 . This completes the proof in this case.□

Let Γ 2 be the family of all connected graphs except for the graphs G , where the vertices in the covering set C 1 of the subgraph B 2 have the property that there are pendent vertices incident to some v ∈ C 1 or any two vertices of C 1 forms a triangle with a vertex u ∈ V ( G ) \ C , where C is the covering set of G .

Theorem 3.2

Let G ∈ Γ 2 be a connected graph of n vertices and m edges with covering number τ . If a book graph B 2 is a subgraph of G, then Brouwer’s conjecture holds for G when k ≥ 2 τ − 1 .

Proof

Let B 2 be a book graph and B 2 ⊆ G . Clearly, τ ( B 2 ) = 2 . Let C = { v 1 , v 2 , … , v τ } be a minimum covering set of G , and C 1 = { v 1 , v 2 } ⊆ C be a minimum covering set of B 2 . Since the Laplacian spectrum of B 2 is { ( t + 2 ) [ 2 ] , 2 [ t − 1 ] , 0 } ,

(6) S k ( B 2 ) = t + 2 , k = 1 , 2 t + 2 k , 2 ≤ k ≤ n − 1 .

By Lemma 2.4, we have S k ( G ) ≤ S k ( B 2 ) + S k ( G \ B 2 ) . In order to establish the theorem, we consider the value of S k ( G \ B 2 ) .

For each i = 3 , 4 , … , τ , let G i be a spanning subgraph of H = G \ B 2 with the edge set E ( G i ) = { u v i ∣ u ∈ N ( v i ) \ { v 1 , v 2 , … , v i − 1 } , v i ∈ C \ { v 1 , v 2 , … , v i − 1 } } and ∣ E ( G i ) ∣ = m i . Clearly, G i = K 1 , m i ∪ ( n ( H ) − m i − 1 ) K 1 , for i = 3 , 4 , … , τ . Note that n ( G i ) = n ( H ) , for each i . Since the Laplacian spectrum of G i is { m i + 1 , 1 [ n ( G i ) − 2 ] , 0 [ n ( H ) − m i ] } , S k ( G i ) ≤ m i + k .

Note that ∑ i = 3 τ m i = m − 2 t − 1 , thus

(7) S k ( H ) ≤ ∑ i = 3 τ S k ( G i ) ≤ ∑ i = 3 τ m i + k = m − 2 t − 1 + ( τ − 2 ) k .

Thus, we have

S k ( G ) ≤ m + τ k − 1 ≤ m + k 2 + k 2 ,

k 2 − ( 2 τ − 1 ) k + 2 ≥ 0 .

Consider the polynomial f ( k ) = k 2 − ( 2 τ − 1 ) k + 2 . The roots of this polynomial are x 1 = ( 2 τ − 1 ) − ( 2 τ − 1 ) 2 − 8 2 and x 2 = ( 2 τ − 1 ) + ( 2 τ − 1 ) 2 − 8 2 . This shows that f ( k ) > 0 , for all k ≥ x 2 and for all k ≤ x 1 . Since τ ( G ) ≥ τ ( B 2 ) = 2 , it is easy to see that x 1 < 1 and x 2 < 2 τ − 1 . This completes the proof.□

Let Γ 3 be the family of all connected graphs except for the graphs G , where the vertices in the covering set C 1 of the subgraph B p have the property that there are pendent vertices incident to some v ∈ C 1 or any two vertices of C 1 forms a triangle with a vertex u ∈ V ( G ) \ C , where C is the covering set of G .

Theorem 3.3

Let G ∈ Γ 3 be a connected graph of n vertices and m edges with covering number τ . If a book graph B p is a subgraph of G, then Brouwer’s conjecture holds for G for k ∈ [ x 1 , n ] when τ ≤ 3 t 2 + 1 , otherwise k ∈ [ y 1 , n ] where x 1 = ( 2 τ − 2 t + 3 ) + ( 2 τ − 2 t + 3 ) 2 − 4 ( 2 t + 6 ) 2 and y 1 = ( 2 τ − 2 t − 1 ) + ( 2 τ − 2 t − 1 ) 2 + 8 t 2 .

Proof

Let B p be a book graph and B p ⊆ G . Clearly, τ ( B p ) = t + 1 . Let C = { v 1 , v 2 , … , v τ } be a minimum covering set of G , and C 1 = { v 1 , v 2 , … , v t + 1 } ⊆ C be a minimum covering set of B p . Since the Laplacian spectrum of B p is { t + 3 , t + 1 , 3 [ t − 1 ] , 2 , 1 [ t − 1 ] , 0 } ,

(8) S k ( B p ) = t + 3 , k = 1 , 2 t + 3 k − 2 , 2 ≤ k ≤ t + 1 . 4 t + k + 1 , t + 2 ≤ k ≤ n − 1 .

By Lemma 2.4, we have S k ( G ) ≤ S k ( B p ) + S k ( G \ B p ) . In order to establish the theorem, we consider the value of S k ( G \ B p ) .

For each i = t + 2 , t + 3 , … , τ , let G i be a spanning subgraph of H = G \ B p with the edge set E ( G i ) = { u v i ∣ u ∈ N ( v i ) \ { v 1 , v 2 , … , v i − 1 } , v i ∈ C \ C 1 } and ∣ E ( G i ) ∣ = m i . Clearly, G i = K 1 , m i ∪ ( n ( H ) − m i − 1 ) K 1 , for i = t + 2 , t + 3 , … , τ . Note that n ( G i ) = n ( H ) , for each i . Since the Laplacian spectrum of G i is { m i + 1 , 1 [ n ( G i ) − 2 ] , 0 [ n ( H ) − m i ] } , S k ( G i ) ≤ m i + k .

Note that ∑ i = t + 2 τ m i = m − 3 t − 1 , thus

(9) S k ( H ) ≤ ∑ i = t + 2 τ S k ( G i ) ≤ ∑ i = t + 2 τ ( m i + k ) = m − 3 t − 1 + ( τ − t − 1 ) k .

For 2 ≤ k ≤ t + 1 , we have

S k ( G ) ≤ m − t − 3 + ( τ − t + 2 ) k ≤ m + k 2 + k 2 ,

k 2 − ( 2 τ − 2 t + 3 ) k + ( 2 t + 6 ) ≥ 0 .

Consider the polynomial f ( k ) = k 2 − ( 2 τ − 2 t + 3 ) k + ( 2 t + 6 ) . The roots of this polynomial are x 1 = ( 2 τ − 2 t + 3 ) + ( 2 τ − 2 t + 3 ) 2 − 4 ( 2 t + 6 ) 2 and x 2 = ( 2 τ − 2 t + 3 ) − ( 2 τ − 2 t + 3 ) 2 − 4 ( 2 t + 6 ) 2 . This shows that f ( k ) ≥ 0 , for all k ≥ x 1 and for all k ≤ x 2 . Since τ ( G ) ≥ τ ( B p ) = 2 , it is easy to see that x 2 < 2 , and x 1 < t + 1 when τ ≤ 3 t 2 + 1 and x 2 ≥ t + 1 when τ ≥ 3 t 2 + 1 .

For t + 2 ≤ k ≤ n − 1 , we have

S k ( G ) ≤ m + t + ( τ − t ) k ≤ m + k 2 + k 2 ,

k 2 − ( 2 τ − 2 t − 1 ) k − 2 t ≥ 0 .

Consider the polynomial f ( k ) = k 2 − ( 2 τ − 2 t − 1 ) k − 2 t . The roots of this polynomial are y 1 = ( 2 τ − 2 t − 1 ) + ( 2 τ − 2 t − 1 ) 2 + 8 t 2 and y 2 = ( 2 τ − 2 t − 1 ) − ( 2 τ − 2 t − 1 ) 2 + 8 t 2 . This shows that f ( k ) > 0 , for all k ≥ y 1 and for all k ≤ y 2 < 0 . It is easy to see that x 4 ≤ t + 2 when τ ≤ 3 t 2 + 1 and x 4 > t + 2 when τ > 3 t 2 + 1 . This completes the proof.□

4 Brouwer’s conjecture for the Halin graph

A Halin graph, denoted by H , is a planar graph constructed as follows: Let T be a tree on at least four vertices. All vertices of T have degree 1 or at least 3. The vertices with degree 1 are called leaves. Let C be a cycle connecting the leaves of T in such a way that C forms the boundary of the unbounded face. Moreover, the vertices of C are called exterior vertices, and the other vertices are called interior vertices. We call K 1 ∨ C n − 1 the wheel graph, denoted by W n . Clearly, W n is the unique Halin graph with only one interior vertex. In particular, we use H c ( t 1 , t 2 ) , H c ( t 1 , t 2 , t 3 ) , and H c ( t 1 , t 2 , t 3 , t 4 ) to denote the Halin graph with two interior vertices, three interior vertices, and four interior vertices, respectively (Figure 2). In this section, we show the truth of Brouwer’s conjecture for the Halin graph.

$Figure 2 Halin graph H c ( t 1 , t 2 ) {H}_{c}\left({t}_{1},{t}_{2}) .$

Figure 2

Halin graph H c ( t 1 , t 2 ) .

Theorem 4.1

Let H be a Halin graph with p interior vertices. Then, Brouwer’s conjecture holds for H.

Proof

Let T n denote the spanning tree of H with V ( T n ) = V ( H ) , and the graph obtained from H by removing the edges of a cycle with n − p vertices, denote the cycle by C n − p . Since the Laplacian spectrum of C n − p is { 2 − 2 cos 2 π i n − p : 1 ≤ i ≤ n − p } and e ( H ) = 2 n − p − 1 . Since e ( H ) = e ( T n ) + ( n − p ) , we have

S k ( H ) ≤ S k ( T n ) + S k ( C n − p ) ≤ 4 k + e ( T n ) + 2 k − 1 = e ( T n ) + 6 k − 1 ≤ e ( H ) + k 2 + k 2 ,

k 2 − 11 k + 2 ( n − p + 1 ) ≥ 0 .

Noting that x = − b 2 a = 5.5 . Thus, if f ( 5 ) ≥ 0 , then f ≥ 0 for all k . It is easy to see that f ( 5 ) ≥ 0 when n − p ≥ 14 .

Next, consider n − p < 14 . Since n ≥ 2 p + 2 , we have p < 12 and n < 26 . With the aid of computer, we can know that Brouwer’s conjecture holds for H for n − p < 14 . This completes the proof.□

5 Brouwer’s conjecture for some classes of graphs

In this section, we show the truth of Brouwer’s conjecture for certain families of graphs. Let G 1 ∼ G 2 denote the graph obtained from G 1 and G 2 by connecting a vertex of G 1 with a vertex of G 2 . Let G 1 ≈ G 2 denote the graph obtained from G 1 and G 2 by inserting two edges between V ( G 1 ) and V ( G 2 ) .

Lemma 5.1

[6] Let G 1 and G 2 be two graphs of order n 1 and n 2 , respectively. If e ( G i ) ≥ 1 and S k i ( G i ) ≤ e ( G i ) + k i + 1 2 for k i = 1 , 2 , … , n i and i = 1 , 2 , then for 1 ≤ k ≤ n 1 + n 2 ,

S k ( G 1 ∼ G 2 ) ≤ e ( G 1 ∼ G 2 ) + k + 1 2 .

Lemma 5.2

[6] Let G 1 and G 2 be two graphs of order n 1 and n 2 , respectively. If e ( G i ) ≥ 2 and S k i ( G i ) ≤ e ( G i ) + k i + 1 2 for k i = 1 , 2 , … , n i and i = 1 , 2 , then for 1 ≤ k ≤ n 1 + n 2 ,

S k ( G 1 ≈ G 2 ) ≤ e ( G 1 ≈ G 2 ) + k + 1 2 .

Let G 1 ⁎ G 2 denote the graph obtained from G 1 and G 2 by inserting p edges between V ( G 1 ) and V ( G 2 ) . An immediate consequence of Lemmas 5.1 and 5.2 is the following corollary.

Corollary 5.3

Let G 1 and G 2 be two graphs with n 1 and n 2 vertices, respectively. If e ( G i ) ≥ p and S k ( G i ) ≤ e ( G i ) + k + 1 2 for k i = 1 , 2 , … , n i and i = 1 , 2 , then S k ( G 1 ⁎ G 2 ) ≤ e ( G 1 ⁎ G 2 ) + k + 1 2 for k ≥ p + 1 .

Let G 1 and G 2 be two graphs on disjoint sets of n 1 and n 2 vertices, respectively. The corona G 1 ∘ G 2 of G 1 and G 2 is defined as the graph obtained by taking one copy of G 1 and n 1 copies of G 2 , and then joining the i th vertex of G 1 to every vertex in the i th copy of G 2 .

Theorem 5.4

Let G 1 = K ω with ω vertices and G 2 = C a with 3 ≤ a ≤ 2 ω − 1 , then Brouwer’s conjecture holds for G 1 ∘ G 2 for all k .

Proof

Consider a connected graph G = G 1 ∘ G 2 with order n = ω ( a + 1 ) and size m . Let F be the graph on a + 1 vertices having all vertices adjacent to a vertex of the clique K ω . Let L ( F ) be the Laplacian matrix of F and let μ 1 ( F ) ≥ μ 2 ( F ) ≥ … ≥ μ a + 1 ( F ) = 0 be the eigenvalues of L ( F ) . Let C ( a + 1 ) × ( a + 1 ) (or simplify C ) be the matrix defined as

(10) C = − 1 0 ⋯ 0 0 0 0 ⋯ 0 0 ⋮ ⋮ ⋯ ⋮ ⋮ 0 0 ⋯ 0 0 0 0 ⋯ 0 0 .

By a suitable labelling of vertices of G , it can be seen that the Laplacian matrix of G can be written as

(11) M = B ⋯ C C ⋮ ⋯ ⋮ ⋮ C ⋯ B C C ⋯ C B ,

where B = L ( F ) − ( ω − 1 ) C . Taking X and β to be a matrix of order zero, B = L ( F ) − ( ω − 1 ) C and C = C ( a + 1 ) × ( a + 1 ) . Using Lemma 2.3, we have σ ( L ( G ) ) = σ ( ω − 1 ) ( L ( F ) − ω C ) ∪ σ ( L ( F ) ) . The eigenvalues of the matrix − ω C are ω with multiplicity one and 0 with multiplicity a . If μ 1 ′ ≥ μ 2 ′ ≥ … ≥ μ a + 1 ′ = 0 are the eigenvalues of the matrix of B − C , then the eigenvalues of the matrix L ( G ) are μ i ′ with multiplicity ω − 1 and μ i , 1 ≤ i ≤ a + 1 . By Lemma 2.4, we know that μ 1 ′ ≤ μ 1 + ω .

By Corollary 5.3, Brouwer’s conjecture holds for k ≥ a + 1 , we need to show that Brouwer’s conjecture holds for k ≤ a . Therefore, for a ≤ ω , it follows that

S k ( G ) ≤ k μ 1 ′ ≤ k ( μ 1 + ω ) ≤ e ( G ) + k ( k + 1 ) 2 = ω ( ω − 1 ) 2 + 2 ω a + k ( k + 1 ) 2 .

If k 2 − ( 2 μ 1 + 2 ω − 1 ) k + ω ( ω − 1 ) + 4 ω a ≥ 0 , the discriminant of the polynomial is Δ = ( 2 μ 1 − 1 ) 2 + 8 ω μ 1 − 16 ω a and its roots are x 1 = ( 2 μ 1 − 1 + 2 ω ) + Δ 2 and x 2 = ( 2 μ 1 − 1 + 2 ω ) − Δ 2 . This shows that f ( k ) ≥ 0 for k ≥ x 1 and k ≤ x 2 . In order to complete the proof, we need to prove that x 2 ≥ a . That is,

x 2 = ( 2 μ 1 + 2 ω − 1 ) − ( 2 μ 1 − 1 ) 2 + 8 ω μ 1 − 16 ω a 2 ≥ a .

Since μ 1 ≤ a + 1 and a ≤ ω , ( 2 μ 1 − 1 ) + 2 ( ω − a ) ≥ ( 2 μ 1 − 1 ) 2 + 8 ω ( μ 1 − 2 a ) always true. This shows that (5.1) holds for all k ≤ a ≤ ω .

For ω + 1 ≤ a ≤ 2 ω − 1 , it follows that

S k ( G ) ≤ ( ω − 1 ) μ 1 ′ + ( k − ω + 1 ) μ 1 ≤ e ( G ) + k ( k + 1 ) 2 = ω ( ω − 1 ) 2 + 2 ω a + k ( k + 1 ) 2 .

If k 2 − ( 2 μ 1 − 1 ) k − ω ( ω − 1 ) + 4 ω a ≥ 0 , the discriminant of the polynomial is Δ = ( 2 μ 1 − 1 ) 2 + 4 ω ( ω − 1 ) − 16 ω a and its roots are x 1 = ( 2 μ 1 − 1 ) + Δ 2 and x 2 = ( 2 μ 1 − 1 ) − Δ 2 . This shows that f ( k ) ≥ 0 for k ≥ x 1 and k ≤ x 2 . In order to complete the proof, we need to prove that x 2 ≥ a . That is,

x 2 = ( 2 μ 1 − 1 ) − ( 2 μ 1 − 1 ) 2 + 4 ω ( ω − 1 ) − 16 ω a 2 ≥ a .

Since μ 1 ≤ a + 1 and ω + 1 ≤ a ≤ 2 ω − 1 , the inequations are always true. This shows that (5.1) holds for all k ≤ a . This completes the proof.□

Acknowledgements

The authors are grateful for the reviewers valuable comments that improved the manuscript.

Funding information: This work was supported by the Science Foundation of Qinghai Province (No. 2021-ZJ-703) and the National Science Foundation of China (Nos. 12261074, 12201335, and 11661068).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript.
Conflict of interest: The authors state no conflict of interest.

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Received: 2024-04-04

Revised: 2024-07-25

Accepted: 2024-08-16

Published Online: 2024-10-22

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2024-0062

Keywords for this article

Brouwer’s conjecture; Laplacian eigenvalues; Halin graph

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