Spectral radius and Hamiltonicity of graphs with large minimum degree (1602.01033v3)

Abstract: This paper presents sufficient conditions for Hamiltonian paths and cycles in graphs. Letting $\lambda\left( G\right) $ denote the spectral radius of the adjacency matrix of a graph $G,$ the main results of the paper are: (1) Let $k\geq1,$ $n\geq k^{3}/2+k+4,$ 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 $G=K_{k}\vee(K_{n-2k}+\overline{K}{k})$. (2) Let $k\geq1,$ $n\geq k^{3}/2+k^{2}/2+k+5,$ 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-2, ] then $G$ has a Hamiltonian path, unless $G=K{k}\vee(K_{n-2k-1}+\overline {K}{k+1})$ or $G=K{n-k-1}+K_{k+1}$ In addition, it is shown that in the above statements, the bounds on $n$ are tight within an additive term not exceeding $2$.