Spectral conditions of pancyclicity for t-tough graphs

TL;DR

This paper investigates spectral conditions that guarantee pancyclicity in t-tough graphs for t in {1, 2, 3}, confirming Bondy's metaconjecture in these cases through spectral graph theory.

Contribution

It establishes spectral criteria involving edge count, spectral radius, and signless Laplacian spectral radius that ensure pancyclicity in t-tough graphs for t in {1, 2, 3}.

Findings

Spectral conditions guarantee pancyclicity for t=1, 2, 3.

Confirms Bondy's conjecture for these t values.

Provides spectral bounds related to graph toughness and cyclic structures.

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.