The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In our recent work, we have determined the graphs with maximal Laplacian spreads among all trees of fixed order and among all unicyclic graphs of fixed order, respectively. In this paper, we continue the work on Laplacian spread of graphs, and prove that there exist exactly two bicyclic graphs with maximal Laplacian spread among all bicyclic graphs of fixed order, which are obtained from a star by adding two incident edges and by adding two nonincident edges between the pendant vertices of the star, respectively.
In this paper, we characterize the trees with the largest Laplacian and adjacency spectral radii among all trees with fixed number of vertices and fixed maximal degree, respectively.
In this paper, an equivalent condition of a graph G with t （2≤ t ≤n） distinct Laplacian eigenvalues is established. By applying this condition to t = 3, if G is regular （necessarily be strongly regular）, an equivalent condition of G being Laplacian integral is given. Also for the case of t = 3, if G is non-regular, it is found that G has diameter 2 and girth at most 5 if G is not a tree. Graph G is characterized in the case of its being triangle-free, bipartite and pentagon-free. In both cases, G is Laplacian integral.
Let G be a mixed glaph which is obtained from an undirected graph by orienting some of its edges. The eigenvalues and eigenvectors of G are, respectively, defined to be those of the Laplacian matrix L（G） of G. As L（G） is positive semidefinite, the singularity of L（G） is determined by its least eigenvalue λ1 （G）. This paper introduces a new parameter edge singularity εs（G） that reflects the singularity of L（G）, which is the minimum number of edges of G whose deletion yields that all the components of the resulting graph are singular. We give some inequalities between εs（G） and λ1 （G） （and other parameters） of G. In the case of εs（G） = 1, we obtain a property on the structure of the eigenvectors of G corresponding to λ1 （G）, which is similar to the property of Fiedler vectors of a simple graph given by Fiedler.
