-common consequents in Boolean matrices
Algebraic conditions for -tough graphs
We give some algebraic conditions for -tough graphs in terms of the Laplacian eigenvalues and adjacency eigenvalues of graphs.
The maximum clique and the signless Laplacian eigenvalues
Lower and upper bounds are obtained for the clique number and the independence number , in terms of the eigenvalues of the signless Laplacian matrix of a graph .
Hall exponents of Boolean matrices
The Laplacian spread of graphs
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 -cyclic graphs with vertices and Laplacian spread are discussed.
On graphs with the largest Laplacian index
Let be a connected simple graph on vertices. The Laplacian index of , namely, the greatest Laplacian eigenvalue of , is well known to be bounded above by . In this paper, we give structural characterizations for graphs with the largest Laplacian index . Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on and for the existence of a -regular graph of order with the largest Laplacian...
The (signless) Laplacian spectral radius of unicyclic and bicyclic graphs with vertices and pendant vertices
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 vertices and pendant vertices is identified. Furthermore, we determine the graphs with the largest Laplacian spectral radii among the all unicyclic graphs and bicyclic graphs with vertices and pendant vertices, respectively....
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