Spectral analogues of Erd\H{o}s' and Moon-Moser's theorems on Hamilton cycles

TL;DR

This paper establishes spectral conditions related to the signless Laplacian for the existence of Hamilton cycles in graphs and bipartite graphs, extending classical degree-based theorems to spectral graph theory.

Contribution

It determines extremal spectral radii for non-Hamiltonian graphs and their complements, generalizing Erd ext{"o}s and Moon-Moser's theorems to spectral parameters.

Findings

Identifies maximum spectral radius for non-Hamiltonian graphs with given minimum degree.

Finds minimum spectral radius of complements of such graphs.

Characterizes extremal graphs achieving these spectral bounds.

Abstract

In 1962, Erd\H{o}s gave a sufficient condition for Hamilton cycles in terms of the vertex number, edge number, and minimum degree of graphs which generalized Ore's theorem. One year later, Moon and Moser gave an analogous result for Hamilton cycles in balanced bipartite graphs. In this paper we present the spectral analogues of Erd\H{o}s' theorem and Moon-Moser's theorem, respectively. Let $\mathcal{G}_n^k$ be the class of non-Hamiltonian graphs of order $n$ and minimum degree at least $k$. We determine the maximum (signless Laplacian) spectral radius of graphs in $\mathcal{G}_n^k$ (for large enough $n$), and the minimum (signless Laplacian) spectral radius of the complements of graphs in $\mathcal{G}_n^k$. All extremal graphs with the maximum (signless Laplacian) spectral radius and with the minimum (signless Laplacian) spectral radius of the complements are determined, respectively. We also solve similar problems for balanced bipartite graphs and the quasi-complements.