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

Strongly Regular Graphs as Laplacian Extremal Graphs