Spectral radius and Hamiltonian properties of graphs

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 $n\ge 4$ vertices with $\delta \ge 1$. 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 $n\ge 14$ vertices with $\delta \ge 2$. If $\rho (G)\ge \rho (K_2\vee (K_{n-4}+2K_1))$, 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.

We fill a gap in the proof of Theorem 2 in previous versions and correct some typos as well

References in corpus (4)

• Spectral analogues of Erdős' and Moon-Moser's theorems on Hamilton cycles
• Spectral analogues of Moon-Moser's theorem on Hamilton paths in bipartite graphs
• Spectral radius and traceability of connected claw-free graphs
• Extremal problems on the Hamiltonicity of claw-free graphs

Cited by in corpus (15)

• Spectral analogues of Erdős' and Moon-Moser's theorems on Hamilton cycles
• The spanning $k$-trees, perfect matchings and spectral radius of graphs
• Spectral analogues of Moon-Moser's theorem on Hamilton paths in bipartite graphs
• Wiener index, Harary index and Hamiltonicity of graphs
• Spectral radius and Hamiltonian properties of graphs, II
• Spectral radius and traceability of connected claw-free graphs
• Signless Laplacian spectral radius and Hamiltonicity of graphs with large minimum degree
• Extremal problems on the Hamiltonicity of claw-free graphs
• The stability method, eigenvalues and cycles of consecutive lengths
• Sufficient spectral conditions for graphs being $k$-edge-Hamiltonian or $k$-Hamiltonian
• A tight $Q$-index condition for a graph to be $k$-path-coverable involving minimum degree
• Spectral Radius and Hamiltonicity of graphs
• Signless Laplacian spectral conditions for Hamilton-connected graphs with large minimum degree
• A note on eigenvalues and Hamiltoinan properties of $k$-connected graphs
• Sufficient spectral conditions on Hamiltonian and traceable graphs