Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees (original) (raw)
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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.
Sharp lower bounds on the Laplacian eigenvalues of trees
Linear Algebra and its Applications, 2004
Let λ 1 (T) and λ 2 (T) be the largest and the second largest Laplacian eigenvalues of a tree T. We obtain the following sharp lower bound for λ 1 (T): λ 1 (T) max (d i + m i + 1) + (d i + m i + 1) 2 − 4(d i m i + 1) 2 : v i ∈ V , where d i and m i denote the degree of vertex v i and the average of the degrees of the vertices adjacent to vertex v i respectively. Equality holds if and only if T is a tree T (d i , d j), where T (d i , d j) is formed by joining the centres of d i copies of K 1,d j −1 to a new vertex v i , that is, T (d i , d j) − v i = d i K 1,d j −1. Let v 1 be the highest degree vertex of degree d 1 and v 2 be the second highest degree vertex of degree d 2. We also show that if T is a tree of order n > 2, then λ 2 (T) d 2 if v 1 v 2 ∈ E, (d 2 +1)+ √ (d 2 +1) 2 −4 2 if v 1 v 2 / ∈ E, where E is the set of edges. Equality holds if T = T 1 (d 1) or T = T 2 (d 1), where T 1 (d 1) is formed by joining the centres of two copies of K 1,d 1 −1 and T 2 (d 1) is formed by joining the centres of two copies of K 1,d 1 −1 to a new vertex. Moreover, we obtain the lower bounds for the sum of two largest Laplacian eigenvalues.
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 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 .
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 .
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
On the second largest Laplacian eigenvalues of graphs
Linear Algebra and its Applications, 2013
The second largest Laplacian eigenvalue of a graph is the second largest eigenvalue of the associated Laplacian matrix. In this paper, we study extremal graphs for the extremal values of the second largest Laplacian eigenvalue and the Laplacian separator of a connected graph, respectively. All simple connected graphs with second largest Laplacian eigenvalue at most 3 are characterized. It is also shown that graphs with second largest Laplacian eigenvalue at most 3 are determined by their Laplacian spectrum. Moreover, the graphs with maximum and the second maximum Laplacian separators among all connected graphs are determined.
On Distribution of Laplacian Eigenvalues of Graphs
2021
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Nonpositive eigenvalues of the adjacency matrix and lower bounds for Laplacian eigenvalues
Discrete Mathematics, 2013
Let NP O(k) be the smallest number n such that the adjacency matrix of any undirected graph with n vertices or more has at least k nonpositive eigenvalues. We show that NP O(k) is well-defined and prove that the values of NP O(k) for k = 1, 2, 3, 4, 5 are 1, 3, 6, 10, 16 respectively. In addition, we prove that for all k ≥ 5, R(k, k + 1) ≥ NP O(k) > T k , in which R(k, k + 1) is the Ramsey number for k and k + 1, and T k is the k th triangular number. This implies new lower bounds for eigenvalues of Laplacian matrices: the k-th largest eigenvalue is bounded from below by the NP O(k)-th largest degree, which generalizes some prior results.