Petals and books: The largest Laplacian spectral gap from 1

Abstract We prove that, for any connected graph on vertices, the spectral gap from the value 1 with respect to the normalized Laplacian is at most 1/2. Moreover, we show that equality is achieved if and only if the graph is either a petal graph (for odd) or a book graph (for even). This implies that is a maximal gap interval for the normalized Laplacian on connected graphs. This is closely related to the Alon–Boppana bound on regular graphs and a recent result by Kollár and Sarnak on cubic graphs. Our result also provides a sharp bound for the convergence rate of some eigenvalues of the Laplacian on neighborhood graphs.