On the second largest Laplacian eigenvalue of trees

Graphs with small second largest Laplacian eigenvalue

Let L(G) be the Laplacian matrix of G. In this paper, we characterize all of the connected graphs with second largest Laplacian eigenvalue no more than l; where l . = 3.2470 is the largest root of the equation μ3 − 5μ2 + 6μ − 1 = 0. Moreover, this result is used to characterize all connected graphs with second largest Laplacian eigenvalue no more than three. © 2013 Elsevier Ltd. All rights rese...

The second largest eigenvalue of trees

Spectral Characterization of Graphs with Small Second Largest Laplacian Eigenvalue

The family G of connected graphs with second largest Laplacian eigenvalue at most θ, where θ = 3.2470 is the largest root of the equation μ−5μ+6μ−1 = 0, is characterized by Wu, Yu and Shu [Y.R. Wu, G.L. Yu and J.L. Shu, Graphs with small second largest Laplacian eigenvalue, European J. Combin. 36 (2014) 190–197]. Let G(a, b, c, d) be a graph with order n = 2a + b + 2c + 3d + 1 that consists of ...

A New Upper Bound on the Largest Normalized Laplacian Eigenvalue

Abstract. Let G be a simple undirected connected graph on n vertices. Suppose that the vertices of G are labelled 1,2, . . . ,n. Let di be the degree of the vertex i. The Randić matrix of G , denoted by R, is the n× n matrix whose (i, j)−entry is 1 √ did j if the vertices i and j are adjacent and 0 otherwise. The normalized Laplacian matrix of G is L = I−R, where I is the n× n identity matrix. ...

On graphs with largest Laplacian eigenvalue at most 4

In this paper graphs with the largest Laplacian eigenvalue at most 4 are characterized. Using this we show that the graphs with the largest Laplacian eigenvalue less than 4 are determined by their Laplacian spectra. Moreover, we prove that ones with no isolated vertex are determined by their adjacency spectra.