Some sufficient conditions on hamilton graphs with toughness

Area of Science:

• Graph Theory
• Spectral Graph Theory
• Combinatorics

Background:

• Graph toughness is a measure of connectivity, defined by the condition tc(G - S) ≤ |S| for vertex cuts S.
• Hamiltonian graphs contain a cycle passing through all vertices, a property studied in relation to graph toughness.
• Previous research by Chvátal and others investigated the link between toughness and cyclic structures in graphs.

Purpose of the Study:

• To establish sufficient conditions for a graph to be Hamiltonian based on its toughness.
• To explore the influence of graph properties like the number of edges, spectral radius, and signless Laplacian spectral radius on Hamiltonicity.

Main Methods:

• The study defines graph toughness and Hamiltonian graphs.
• It analyzes the relationship between these properties using graph-theoretic concepts.
• The research focuses on sufficient conditions derived from spectral properties and edge counts.

Main Results:

• New sufficient conditions are established for graphs to possess Hamiltonian cycles.
• These conditions integrate the concepts of graph toughness, spectral radius, and signless Laplacian spectral radius.
• The findings provide insights into the structural properties that guarantee Hamiltonicity.

Conclusions:

• The paper contributes to understanding the interplay between graph toughness and Hamiltonicity.
• The established conditions offer a framework for identifying Hamiltonian graphs based on spectral and edge characteristics.
• This research advances the study of spectral graph theory and its applications in determining graph properties.