Bounds for Laplacian graph eigenvalues

Bounds for Laplacian Graph Eigenvalues

Let G be a connected simple graph whose Laplacian eigenvalues are 0 = μn (G) μn−1 (G) · · · μ1 (G) . In this paper, we establish some upper and lower bounds for the algebraic connectivity and the largest Laplacian eigenvalue of G . Mathematics subject classification (2010): 05C50, 15A18.

Sharp upper bounds for the Laplacian graph eigenvalues

Let G = (V ,E) be a simple connected graph and λ1(G) be the largest Laplacian eigenvalue of G. In this paper, we prove that: 1. λ1(G) = max{du +mu : u ∈ V } if and only if G is a regular bipartite or a semiregular bipartite graph, where du and mu denote the degree of u and the average of the degrees of the vertices adjacent to u, respectively. 2. λ1(G) = 2 + √ (r − 2)(s − 2) if and only if G is...

Majorization bounds for signless Laplacian eigenvalues

Graph Embeddings and Laplacian Eigenvalues

Graph embeddings are useful in bounding the smallest nontrivial eigenvalues of Laplacian matrices from below. For an n×n Laplacian, these embedding methods can be characterized as follows: The lower bound is based on a clique embedding into the underlying graph of the Laplacian. An embedding can be represented by a matrix Γ; the best possible bound based on this embedding is n/λmax(Γ Γ). Howeve...

Bounds on normalized Laplacian eigenvalues of graphs

*Correspondence: [email protected] 1School of Mathematics and Statistics, Minnan Normal University, Zhangzhou, Fujian, P.R. China 2Center for Discrete Mathematics, Fuzhou University, Fuzhou, Fujian, P.R. China Full list of author information is available at the end of the article Abstract Let G be a simple connected graph of order n, where n≥ 2. Its normalized Laplacian eigenvalues are 0 = λ1 ...