Graph theory, the study of mathematical structures made of points called vertices connected by lines known as edges, has long been an important field in mathematics. Graph theory is a mathematically interesting subject and has a wide range of applications in different fields such as computer science, chemistry, physics, biology, social sciences, and etc. 

One important concept in graph theory is the degree of a vertex in a graph which is defined as the number of edges incident to that vertex. The first Zagreb index of a graph is defined as the sum of the squares of the degrees of the vertices in a graph. The first Zagreb index was introduced by Gutman and Trinajstić in 1972 and was rooted from the study of chemical graph theory in which the degree of each vertex in a graph is less than or equal to four. The first Zagreb index is one of the most important topological indexes of a graph and has been investigated intensively for many years. 

A graph is called a Hamiltonian graph if the graph has a cycle containing all vertices in the graph. The Hamilton problem in graph theory is to find a characterization of a Hamiltonian graph. Mathematically, it is to find a condition which is sufficient and necessary for a Hamiltonian graph. The Hamilton problem is one major unsolved problem in graph theory. While investigating the Hamilton problem, the investigators often focus on finding the sufficient conditions for a Hamiltonian graph. 

Recently, Professor Rao Li from the University of South Carolina Aiken presented new sufficient conditions based on the first Zagreb index for a Hamiltonian graph. The research has been published in the peer-reviewed journal Mathematics. During the research, Professor Li utilized the well-known Chvátal-Erdös theorem in Hamiltonian graph theory, one observation on a graph, and two inequalities established by Shisha and Mond in 1967. 

It is commonly believed that it is hard to find a closed mathematical expression for the first Zagreb index of a graph. The researchers often focus on obtaining the bounds of the first Zagreb index. Professor Li realized that the ideas and techniques developed in obtaining the sufficient conditions for a Hamiltonian graph can be employed to establish new upper bounds for the first Zagreb index. After performing careful analyses, Professor Li eventually presented two new achievable upper bounds for the first Zagreb index in the same paper.

“It is very interesting to see we are able to use the inequalities in mathematical analysis to find new sufficient conditions involving the first Zagreb index for a Hamiltonian graph and new upper bounds for the first Zagreb index of a graph. This research shows the new applications of the first Zagreb index and enriches the studies on Hamiltonian graph theory and the first Zagreb index of a graph.” said Professor Li.

Journal Reference

Li, R. “The First Zagreb Index and Some Hamiltonian Properties of Graphs.” Mathematics, 2024. DOI: https://doi.org/10.3390/math12243902