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Eigenvalues and cycles of consecutive lengths

Binlong Li, Bo Ning

Abstract

As 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 $C$ such that for all positive $\varepsilon<C$ and sufficiently large $n$, every graph $G$ of order $n$ with spectral radius $ρ(G)>\sqrt{\lfloor\frac{n^2}{4}\rfloor}$ 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, 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.

Eigenvalues and cycles of consecutive lengths

Abstract

As 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 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, 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.

Paper Structure

This paper contains 1 section, 10 theorems, 8 equations.

Table of Contents

  1. Acknowledgment

Key Result

Theorem 1

If $0<\varepsilon<10^{-6}$, then we can choose $N=2.5\times 10^{10}{\varepsilon}^{-1}$. Let $\varepsilon$ be real with $0<\varepsilon<\frac{1}{4}$. Then there exists an integer $N:=N(\varepsilon)$, such that if $G$ is a graph on $n$ vertices with $n\geq N$ and $\rho(G)>\sqrt{\lfloor\frac{n^2}{4}\rfl

Theorems & Definitions (12)

  • Theorem 1
  • Theorem A: Voss and Zuluaga VZ77
  • Theorem B: Ore O61
  • Theorem C: Gould, Haxell and Scott GHS02
  • Theorem D: Hong H93
  • Theorem E: Sun and Das SD20
  • Theorem F: Nosal N70
  • Lemma 1
  • Lemma 2
  • proof
  • ...and 2 more