  The Laplacian spread of a graph is the difference between the largest
eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the
graph. We find that the class of strongly regular graphs attains the maximum of
largest eigenvalues, the minimum of second-smallest eigenvalues of Laplacian
matrices and hence the maximum of Laplacian spreads among all simple connected
graphs of fixed order, minimum degree, maximum degree, minimum size of common
neighbors of two adjacent vertices and minimum size of common neighbors of two
nonadjacent vertices. Some other extremal graphs are also provided.
