Spectral conditions of pancyclicity for t-tough graphs

Abstract: More than 40 years ago Chv\'atal introduced a new graph invariant, which he called graph toughness. From then on a lot of research has been conducted, mainly related to the relationship between toughness conditions and the existence of cyclic structures, in particular, determining whether the graph is Hamiltonian and pancyclic. A pancyclic graph is certainly Hamiltonian, but not conversely. Bondy in 1976, however, 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. We confirm the Bondy conjecture for t-tough graphs in the case when $t\in { 1;2;3}$ in terms of the edge number, the spectral radius and the signless Laplacian spectral radius of the graph.