Eigenvalues and cycles of consecutive lengths

TL;DR

This paper investigates the relationship between spectral radius and cycle lengths in graphs, establishing a new lower bound for the maximum cycle length guaranteed by spectral conditions, and introduces novel proof techniques.

Contribution

It improves the bound on the maximum cycle length in graphs with large spectral radius and develops new methods inspired by recent research on Ramsey numbers and spectral inequalities.

Findings

Proves that C ≥ 1/4 for large graphs with spectral radius above a threshold.

Introduces a novel proof technique combining spectral inequalities and ideas from Ramsey theory.

Derives an Erdős-Gallai-type condition for even cycles, of independent interest.

Abstract

As the counterpart of classical theorems on cycles of consecutive lengths due to Bondy and Bollob\'as in spectral graph theory, Nikiforov proposed the following open problem in 2008: What is the maximum $C$ such that for all positive $\varepsilon<C$ and sufficiently large $n$, every graph $G$ of order $n$ with spectral radius $\rho(G)>\sqrt{\lfloor\frac{n^2}{4}\rfloor}$ contains a cycle of length $\ell$ for each integer $\ell\in[3,(C-\varepsilon)n]$. We prove that $C\geq\frac{1}{4}$ by a novel method, improving the existing bounds. Besides several novel ideas, our proof technique is partly inspirited by the recent research on Ramsey numbers of star versus large even cycles due to Allen, {\L}uczak, Polcyn and Zhang, and with aid of a powerful spectral inequality. We also derive an Erd\H{o}s-Gallai-type edge number condition for even cycles, which may be of independent interest.