On the sum of the Laplacian eigenvalues of a graph and Brouwer's conjecture (original) (raw)
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Upper bounds for the sum of Laplacian eigenvalues of graphs
Linear Algebra and its Applications, 2012
Let G be a graph with n vertices and e(G) edges, and let μ 1 (G) ≥ μ 2 (G) ≥ • • • ≥ μ n (G) = 0 be the Laplacian eigenvalues of G. Let S k (G) = k i=1 μ i (G), where 1 ≤ k ≤ n. Brouwer conjectured that S k (G) ≤ e(G) + k+1 2 for 1 ≤ k ≤ n. It has been shown in Haemers et al. [7] that the conjecture is true for trees. We give upper bounds for S k (G), and in particular, we show that the conjecture is true for unicyclic and bicyclic graphs.
On the sum of Laplacian eigenvalues of graphs
Linear Algebra and its Applications, 2010
Let k be a natural number and let G be a graph with at least k vertices. A.E. Brouwer conjectured that the sum of the k largest Laplacian eigenvalues of G is at most e(G) + k+1 2 , where e(G) is the number of edges of G. We prove this conjecture for k = 2. We also show that if G is a tree, then the sum of the k largest Laplacian eigenvalues of G is at most e(G) + 2k − 1.
Linear Algebra and its Applications, 2004
Let G = (V , E) be a simple connected graph and λ 1 (G) be the largest Laplacian eigenvalue of G. In this paper, we prove that: 1. λ 1 (G) = d 1 + d 2 , (d 1 / = d 2) if and only if G is a star graph, where d 1 , d 2 are the highest and the second highest degree, respectively. 2. λ 1 (G) = max 2(d 2 u + d u m u) : u ∈ V if and only if G is a bipartite regular graph, where m u = v {d v −|N u ∩N v |:uv∈E} d u , d u denotes the degree of u and |N u ∩ N v | is the number of common neighbors of u and v. 3. λ 1 (G) max (d u +d v)+ √ (d u −d v) 2 +4m u m v 2 : uv ∈ E with equality if and only if G is a bipartite regular graph or a bipartite semiregular graph, where d u and m u denote the degree of u and the average of the degrees of the vertices adjacent to u, respectively.
On the Laplacian eigenvalues of a graph and Laplacian energy
Linear Algebra and its Applications, 2015
Let G be a simple graph with n vertices, m edges, maximum degree Δ, average degree d = 2m n , clique number ω having Laplacian eigenvalues μ 1 , μ 2 ,. .. , μ n−1 , μ n = 0. For k (1 ≤ k ≤ n), let S k (G) = k i=1 μ i and let σ (1 ≤ σ ≤ n − 1) be the number of Laplacian eigenvalues greater than or equal to average degree d. In this paper, we obtain a lower bound for S ω−1 (G) and an upper bound for S σ (G) in terms of m, Δ, σ and clique number ω of the graph. As an application, we obtain the stronger bounds for the Laplacian energy LE(G) = n i=1 |μ i − d|, which improve some well known earlier bounds.
An improved upper bound for Laplacian graph eigenvalues
Linear Algebra and its Applications, 2003
Let G = (V , E) be a simple graph on vertex set V = {v 1 , v 2 ,. .. , v n }. Further let d i be the degree of v i and N i be the set of neighbors of v i. It is shown that max d i + d j − |N i ∩ N j | : 1 i < j n, v i v j ∈ E is an upper bound for the largest eigenvalue of the Laplacian matrix of G, where |N i ∩ N j | denotes the number of common neighbors between v i and v j. For any G, this bound does not exceed the order of G. Further using the concept of common neighbors another upper bound for the largest eigenvalue of the Laplacian matrix of a graph has been obtained as max 2 d 2 i + d i m i : 1 i n , where m i = j d j − |N i ∩ N j | : v i v j ∈ E d i .
Upper bound for the Laplacian eigenvalues of a graph
2011
In this note we give a new upper bound for the Laplacian eigenvalues of an unweighted graph. Let GGG be a simple graph on nnn vertices. Let dm(G)d_{m}(G)dm(G) and lambdam+1(G)\lambda_{m+1}(G)lambdam+1(G) be the mmm-th smallest degree of GGG and the m+1m+1m+1-th smallest Laplacian eigenvalue of GGG respectively. Then $ \lambda_{m+1}(G)\leq d_{m}(G)+m-1 $ for barGneqKm+(n−m)K1\bar{G} \neq K_{m}+(n-m)K_1 barGneqK_m+(n−m)K_1. We also introduce upper and lower bound for the Laplacian eigenvalues of weighted graphs, and compare it with the special case of unweighted graphs.
Some remarks on Laplacian eigenvalues and Laplacian energy of graphs
2010
Sažetak Suppose mu1\ mu_1 mu1, mu2\ mu_2 mu2,..., mun\ mu_n mun are Laplacian eigenvalues of a graph $ G .TheLaplacianenergyof. The Laplacian energy of .TheLaplacianenergyof G $ is defined as $ LE (G)=\ sum_ {i= 1}^ n|\ mu_i-2m/n| .Inthispaper,somenewboundsfortheLaplacianeigenvaluesandLaplacianenergyofsomespecialtypesofthesubgraphsof. In this paper, some new bounds for the Laplacian eigenvalues and Laplacian energy of some special types of the subgraphs of .Inthispaper,somenewboundsfortheLaplacianeigenvaluesandLaplacianenergyofsomespecialtypesofthesubgraphsof K_n $ are presented.
Some Lower Bounds for Laplacian Energy of Graphs
2009
The Laplacian energy of a graph G is defined as LE(G )= n=1 |λi − 2m n |, where λ1(G) ≥ λ2(G), ..., ≥ λn(G) = 0 are the Laplacian eigenvalues of the graph G. Some lower bounds for Laplacian energy of graphs are presented in this note.