Sufficient conditions for $t$-tough graphs to be Hamiltonian and pancyclic or bipartite

The toughness of graph $G$, denoted by $\tau (G)$, is $\tau (G)=\min \{\frac{\mathrm{\mid }S\mathrm{\mid }}{c(G-S)}:S\subseteq V(G),c(G-S)\ge 2\}$ for every vertex cut $S$ of $V(G)$ and the number of components of $G$ is denoted by $c(G)$. Bondy in 1973, suggested the ``metaconjecture" that almost any nontrivial condition on a graph which implies that the graph is Hamiltonian also implies that the graph is pancyclic. Recently, Benediktovich [Discrete Applied Mathematics. 365 (2025) 130–137] confirmed the Bondy’s metaconjecture for $t$-tough graphs in the case when $t\in \{1;2;3\}$ in terms of the size, the spectral radius and the signless Laplacian spectral radius of the graph. In this paper, we will confirm the Bondy’s metaconjecture for $t$-tough graphs in the case when $t\ge 4$ in terms of the size, the spectral radius, the signless Laplacian spectral radius, the distance spectral radius and the distance signless Laplacian spectral radius of graphs.