Bounding the gap between extremal Laplacian eigenvalues of graphs (original) (raw)
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Bounds on normalized Laplacian eigenvalues of graphs
Journal of Inequalities and Applications, 2014
Let G be a simple connected graph of order n, where n ≥ 2. Its normalized Laplacian eigenvalues are 0 = λ 1 ≤ λ 2 ≤ · · · ≤ λ n ≤ 2. In this paper, some new upper and lower bounds on λ n are obtained, respectively. Moreover, connected graphs with λ 2 = 1 (or λ n-1 = 1) are also characterized.
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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.
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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...
Improved Bounds for the Extremal non-trivial Laplacian Eigenvalues
Gazi University Journal of Science, 2015
Let ܩ be a simple connected graph and its Laplacian eigenvalues be ߤ ଵ ߤ ଶ ⋯ ߤ ିଵ ߤ ൌ 0. In this paper, we present an upper bound for the algebraic connectivity ߤ ିଵ of ܩ and a lower bound for the largest eigenvalue ߤ ଵ of ܩ in terms of the degree sequence ݀ ଵ , ݀ ଶ ,. .. , ݀ of ܩ and the number |ܰ ∩ ܰ | of common vertices of ݅ and ݆ ሺ1 ݅ ൏ ݆ ݊ሻ and hence we improve bounds of Maden and Büyükköse [14].
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.
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 Laplacian eigenvalues of connected graphs
Czechoslovak Mathematical Journal, 2015
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Average distance in graphs and eigenvalues
Discrete Mathematics, 2009
Brendan McKay gave the following formula relating the average distance between pairs of vertices in a tree T and the eigenvalues of its Laplacian: d T = 2 n − 1 n i=2 1 λ i. By modifying Mohar's proof of this result, we prove that for any graph G, its average distance, d G , between pairs of vertices satisfies the following inequality: d G ≥ 2 n − 1 n i=2 1 λ i. This solves a conjecture of Graffiti. We also present a generalization of this result to the average of suitably defined distances for k subsets of a graph.
A lower bound for the second largest Laplacian eigenvalue of weighted graphs
Electronic Journal of Linear Algebra, 2011
Let G be a weighted graph on n vertices. Let λ n−1 (G) be the second largest eigenvalue of the Laplacian of G. For n ≥ 3, it is proved that λ n−1 (G) ≥ d n−2 (G), where d n−2 (G) is the third largest degree of G. An upper bound for the second smallest eigenvalue of the signless Laplacian of G is also obtained.