Spectral conditions of pancyclicity for t-tough graphs

Abstract

More than 50 years ago Chvátal 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 1973, 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’s metaconjecture 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.