Sharp lower bounds on the Laplacian eigenvalues of trees (original) (raw)
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Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees
Discrete Mathematics, 2005
Let G be a simple graph, its Laplacian matrix is the difference of the diagonal matrix of its degrees and its adjacency matrix. Denote its eigenvalues by (G) = 1 (G) 2 (G) · · · n (G) = 0. A vertex of degree one is called a pendant vertex. Let T n,k be a tree with n vertices, which is obtained by adding paths P 1 , P 2 , . . . , P k of almost equal the number of its vertices to the pendant vertices of the star K 1,k . In this paper, the following results are given:
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 .
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.
A Sharp Upper Bound for the Largest Eigenvalue of the Laplacian Matrix of a Tree
2011
This paper contains results in the field of algebraic graph theory and specifically concerns the spectral radius of the Laplacian matrix of a tree. Let A(G) denote the adjacency matrix of a simple graph G. Then, the Laplacian matrix of G is given by L(G) = D(G) − A(G) where D is the diagonal matrix whose diagonal entries are the vertex degrees. The main result provides an upper bound for the spectral radius of any tree with n vertices and k pendant vertices.
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.
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 .
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.
An interlacing approach for bounding the sum of Laplacian eigenvalues of graphs
Linear Algebra and its Applications, 2014
We apply eigenvalue interlacing techniques for obtaining lower and upper bounds for the sums of Laplacian eigenvalues of graphs, and characterize equality. This leads to generalizations of, and variations on theorems by Grone, and Grone & Merris. As a consequence we obtain inequalities involving bounds for some well-known parameters of a graph, such as edge-connectivity, and the isoperimetric number.
Largest Laplacian Eigenvalue and Degree Sequences of Trees
2008
We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the vertices that is obtained by breadth-first search. This structure is uniquely determined up to isomorphism. We also show that the maximum