Toughness, Hamiltonicity and eigenvalues of graphs

For a real number $t\ge 0$, we say a graph $G=(V,E)$ is t-tough if $\left|S\right|\ge t\cdot c(G-S)$ for all $S\subseteq V\left(G\right)$ with $c(G-S)\ge 2$, where $c(G-S)$ is the number of components of $G-S$. The toughness $\tau \left(G\right)$ of G is the maximum t for which G is t-tough. Firstly, we provide a lower bound for $\tau \left(G\right)$ in terms of its normalized Laplacian eigenvalues, improving or generalizing known lower bounds established by Huang, Das and Zhu (2022), Gu (2021) and Zhang (2023). We also derive upper bounds for certain eigenvalues in a regular graph to ensure that the graph is t-tough, where $\frac{1}{t}$ is an integer, which extends the related result of Cioabă and Wong (2014). Additionally, we establish a sufficient condition involving the number of r-cliques to ensure the existence of a Hamiltonian cycle in a t-tough graph, where r is an integer with $r\ge 2$, which improves upon the sufficient condition based on the number of edges proposed by Cai, Yu, Xu and Yu (2022). Finally, we provide a spectral condition to guarantee the existence of a Hamiltonian cycle in t-tough graphs, thereby addressing the problem posed by Fan, Lin and Lu (2023) for integers $t\ge 1$.