Spectral radius and Hamiltonian properties of graphs

TL;DR

This paper establishes spectral radius conditions that guarantee the existence of Hamiltonian paths and cycles in graphs, refining and extending previous spectral graph theory results.

Contribution

It provides new spectral radius criteria for Hamiltonian properties, including characterizations and refinements of earlier theorems.

Findings

Graphs with spectral radius > n-3 contain Hamilton paths unless specific exceptions.

Graphs with spectral radius ≥ that of a particular join graph contain Hamilton cycles unless equal to that graph.

Refines previous theorems by Fiedler, Nikiforov, and Lu et al. on spectral conditions for Hamiltonicity.

Abstract

Let $G$ be a graph with minimum degree $\delta$. The spectral radius of $G$, denoted by $\rho(G)$, is the largest eigenvalue of the adjacency matrix of $G$. In this note we mainly prove the following two results. (1) Let $G$ be a graph on $n\geq 4$ vertices with $\delta\geq 1$. If $\rho(G)> n-3$, then $G$ contains a Hamilton path unless $G\in\{K_1\vee (K_{n-3}+2K_1),K_2\vee 4K_1,K_1\vee (K_{1,3}+K_1)\}$. (2) Let $G$ be a graph on $n\geq 14$ vertices with $\delta \geq 2$. If $\rho(G)\geq \rho(K_2\vee (K_{n-4}+2K_1))$, then $G$ contains a Hamilton cycle unless $G= K_2\vee (K_{n-4}+2K_1)$. As corollaries of our first result, two previous theorems due to Fiedler and Nikiforov and Lu et al. are obtained, respectively. Our second result refines another previous theorem of Fiedler and Nikiforov.