Spectral radius and Hamiltonian properties of graphs

Abstract

Let $G$ be a graph with minimum degree $\hat{I}´$. The spectral radius of $G$, denoted by $\ddot{I}(G)$, 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 $\hat{I}´\ge 1$. If $\ddot{I}(G)>n-3$, 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 $\hat{I}´\ge 2$. If $\ddot{I}(G)\ge \ddot{I}(K_2\vee (K_{n-4}+2K_1))$, 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