Imagine you have a web of interconnected cities, and you want to find the best route that visits every city exactly once. Well, mathematicians have been interested in a similar problem for graphs, called Hamiltonian cycles. In this article, researchers delve into the connection between the toughness of a graph and its ability to have a special cycle that passes through all its vertices. By investigating properties such as the number of edges, spectral radius, and signless Laplacian spectral radius of the graph, they establish conditions that ensure a graph with toughness is Hamiltonian. This research sheds light on the intricate interplay between these two concepts in the world of graphs.
Let G be a graph, and the number of components of G is denoted by c(G). Let t be a positive real number. A connected graph G is t-tough if tc(G − S) ≤ |S| for every vertex cut S of V(G). The toughness of G is the largest value of t for which G is t-tough, denoted by τ(G). We call a graph G Hamiltonian if it has a cycle that contains all vertices of G. Chvátal and other scholars investigate the relationship between toughness conditions and the existence of cyclic structures. In this paper, we establish some sufficient conditions that a graph with toughness is Hamiltonian based on the number of edges, spectral radius, and signless Laplacian spectral radius of the graph.MR subject classifications: 05C50, 15A18.
Dr. David Lowemann, M.Sc, Ph.D., is a co-founder of the Institute for the Future of Human Potential, where he leads the charge in pioneering Self-Enhancement Science for the Success of Society. With a keen interest in exploring the untapped potential of the human mind, Dr. Lowemann has dedicated his career to pushing the boundaries of human capabilities and understanding.
Armed with a Master of Science degree and a Ph.D. in his field, Dr. Lowemann has consistently been at the forefront of research and innovation, delving into ways to optimize human performance, cognition, and overall well-being. His work at the Institute revolves around a profound commitment to harnessing cutting-edge science and technology to help individuals lead more fulfilling and intelligent lives.
Dr. Lowemann’s influence extends to the educational platform BetterSmarter.me, where he shares his insights, findings, and personal development strategies with a broader audience. His ongoing mission is shaping the way we perceive and leverage the vast capacities of the human mind, offering invaluable contributions to society’s overall success and collective well-being.