On the sum of Laplacian eigenvalues of graphs (original) (raw)
On the sum of the Laplacian eigenvalues of a graph and Brouwer's conjecture
Linear Algebra and its Applications, 2016
For a simple graph G with n-vertices, m edges and having Laplacian eigenvalues μ 1 , μ 2 ,. .. , μ n−1 , μ n = 0, let S k (G) = k i=1 μ i , be the sum of k largest Laplacian eigenvalues of G. Brouwer conjectured that S k (G) ≤ m + k+1 2 , for all k = 1, 2,. .. , n. We obtain upper bounds for S k (G) in terms of the clique number ω, the vertex covering number τ and the diameter d of a graph G. We show that Brouwer's conjecture holds for certain classes of graphs. The Laplacian energy LE(G) of a graph G is defined as LE(G) = n i=1 |μ i − d|, where d = 2m n is the average degree of G. We obtain an upper bound for the Laplacian energy LE(G) of a graph G in terms of the number of vertices n, the number of edges m, the vertex covering number τ and the clique number ω of the graph.
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 Distribution of Laplacian Eigenvalues of Graphs
2021
The work in this thesis concerns the investigation of eigenvalues of the Laplacian matrix, normalized Laplacian matrix, signless Laplacian matrix and distance signless Laplacian matrix of graphs. In Chapter 1, we present a brief introduction of spectral graph theory with some definitions. Chapter 222 deals with the sum of $ k $ largest Laplacian eigenvalues $ S_{k}(G) $ of graph $ G $ and Brouwer's conjecture. We obtain the upper bounds for $ S_{k}(G) $ for some classes of graphs and use them to verify Brouwer's conjecture for these classes of graphs. Also, we prove Brouwer's conjecture for more general classes of graphs. In Chapter 333, we investigate the Laplacian eigenvalues of graphs and the Laplacian energy conjecture for trees. We prove the Laplacian energy conjecture completely for trees of diameter $ 4 .Further,weprovethisconjectureforalltreeshavingatmost. Further, we prove this conjecture for all trees having at most .Further,weprovethisconjectureforalltreeshavingatmost \frac{9n}{25}-2 $ non-pendent vertices. Also, we obtain the sufficient conditions for the truth...
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 sum of powers of the Laplacian eigenvalues of graphs
Linear Algebra and its Applications, 2008
For a graph G and a real α / = 0, we study the graph invariant s α (G)-the sum of the αth power of the non-zero Laplacian eigenvalues of G. The cases α = 2, 1 2 and −1 have appeared in different problems. Here we establish some properties for s α with α / = 0, 1. We also discuss the cases α = 2, 1 2 .
A note on sum of powers of the Laplacian eigenvalues of graphs
Applied Mathematics Letters, 2011
For a graph G and a real number α ̸ = 0, the graph invariant s α (G) is the sum of the αth power of the non-zero Laplacian eigenvalues of G. This note presents some bounds for s α (G) in terms of the vertex degrees of G, and a relation between s α (G) and the first general Zagreb index, which is a useful topological index and has important applications in chemistry.
On the sum of the Laplacian eigenvalues of a tree
Linear Algebra and Its Applications, 2011
Given an n-vertex graph G = (V, E), the Laplacian spectrum of G is the set of eigenvalues of the Laplacian matrix L = D − A, where D and A denote the diagonal matrix of vertex-degrees and the adjacency matrix of G, respectively. In this paper, we study the Laplacian spectrum of trees. More precisely, we find a new upper bound on the sum of the k largest Laplacian eigenvalues of every n-vertex tree, where k ∈ {1, . . . , n}. This result is used to establish that the n-vertex star has the highest Laplacian energy over all n-vertex trees, which answers affirmatively to a question raised by Radenković and Gutman .