Bounding the diameter and the mean distance of a graph from its eigenvalues: Laplacian versus adjacency matrix methods Original Research Article

Recently, several results bounding above the diameter and/or the mean distance of a graph from its eigenvalues have been presented. They use the eigenvalues of either the adjacency or the Laplacian matrix of the graph. The main object of this paper is to compare both methods. As expected, they are equivalent for regular graphs. However, the situation is different for nonregular graphs: While no method has a definite advantage when bounding above the diameter, the use of the Laplacian matrix seems better when dealing with the mean distance. This last statement follows from improved bounds on the mean distance obtained in the paper.