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The Laplacian spread of graphs

Zhifu YouBo Lian Liu — 2012

Czechoslovak Mathematical Journal

The Laplacian spread of a graph is defined as the difference between the largest and second smallest eigenvalues of the Laplacian matrix of the graph. In this paper, bounds are obtained for the Laplacian spread of graphs. By the Laplacian spread, several upper bounds of the Nordhaus-Gaddum type of Laplacian eigenvalues are improved. Some operations on Laplacian spread are presented. Connected c -cyclic graphs with n vertices and Laplacian spread n - 1 are discussed.

On graphs with the largest Laplacian index

Bo Lian LiuZhibo ChenMuhuo Liu — 2008

Czechoslovak Mathematical Journal

Let G be a connected simple graph on n vertices. The Laplacian index of G , namely, the greatest Laplacian eigenvalue of G , is well known to be bounded above by n . In this paper, we give structural characterizations for graphs G with the largest Laplacian index n . Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on n and k for the existence of a k -regular graph G of order n with the largest Laplacian...

The (signless) Laplacian spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices

Muhuo LiuXuezhong TanBo Lian Liu — 2010

Czechoslovak Mathematical Journal

In this paper, the effects on the signless Laplacian spectral radius of a graph are studied when some operations, such as edge moving, edge subdividing, are applied to the graph. Moreover, the largest signless Laplacian spectral radius among the all unicyclic graphs with n vertices and k pendant vertices is identified. Furthermore, we determine the graphs with the largest Laplacian spectral radii among the all unicyclic graphs and bicyclic graphs with n vertices and k pendant vertices, respectively....

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