On the Spectrum, the Growth, and the Diameter of a Graph

A lower bound is given for the harmonic mean of the growth in a finite undirected graphΓin terms of the eigenvalues of the Laplacian ofΓ. For a connected graph, this bound is tight if and only if the graph is distance-regular. Bounds on the diameter of a “sphere-regular” graph follow. Finally, a lower bound is given for the growth in an infinite undirected graph of bounded degree in terms of the spectrum of its Laplacian.