Spectral radius and Hamiltonian properties of graphs, II

TL;DR

This paper establishes spectral conditions for Hamiltonian properties in graphs, including cycles and bipartite graphs, extending classical theorems and proposing new spectral criteria for long cycles.

Contribution

It provides new spectral conditions for Hamiltonian cycles and long cycles in graphs, extending classical results and conjectures in spectral graph theory.

Findings

Spectral conditions for the existence of $C_{n-1}$ in 2-connected graphs.

Spectral criteria for Hamilton cycles in balanced bipartite graphs.

Extension of Moon-Moser's theorem using spectral methods.

Abstract

In this paper, we first present spectral conditions for the existence of $C_{n-1}$ in graphs (2-connected graphs) of order $n$, which are motivated by a conjecture of Erd\H{o}s. Then we prove spectral conditions for the existence of Hamilton cycles in balanced bipartite graphs. This result presents a spectral analog of Moon-Moser's theorem on Hamilton cycles in balanced bipartite graphs, and extends a previous theorem due to Li and the second author for $n$ sufficiently large. We conclude this paper with two problems on tight spectral conditions for the existence of long cycles of given lengths.