Eigenvalues and cycles of consecutive lengths

Abstract

AbstractAs the counterpart of classical theorems on cycles of consecutive lengths due to Bondy and Bollobás in spectral graph theory, Nikiforov proposed the following open problem in 2008: What is the maximum such that for all positive and sufficiently large , every graph of order with spectral radius contains a cycle of length for each integer . We prove that , improving the existing bounds. Besides several novel ideas, our proof technique is partly inspired by the recent research on Ramsey numbers of star versus large even cycles due to Allen, Łuczak, Polcyn, and Zhang, and with aid of a powerful spectral inequality. We also derive an Erdős–Gallai‐type edge number condition for even cycles, which may be of independent interest.