Spectral radius and Hamiltonicity of graphs with large minimum degree

Vladimir Nikiforov

Published 2016-02-02Version 1

We extend some recent results on sufficient conditions for Hamiltonian paths and cycles in $G$. \ Let $G$ be a graph of order $n$ and $\lambda\left( G\right) $ be the spectral radius of its adjacency matrix. One of the main results of the paper is the following theorem: Let $k\geq2,$ $n\geq k^{3}+k+4,$ and let $G$ be a graph of order $n$, with minimum degree $\delta\left( G\right) \geq k.$ If \[ \lambda\left( G\right) \geq n-k-1, \] then $G$ has a Hamiltonian cycle, unless $G=K_{1}\vee(K_{n-k-1}+K_{k})$ or $G=K_{k}\vee(K_{n-2k}+\bar{K}_{k})$.