Graph theory is the study of mathematical structures, which are composed of the number of points called edge-connected points and have long been an important field of mathematics. Graph theory is a mathematically interesting topic with wide application in different fields such as computer science, chemistry, physics, biology, social sciences, etc.
An important concept in graph theory is the degree of vertices in a graph, which is defined as the number of edges of that vertex. The first Zagreb index of the graph is defined as the sum of squares of the vertex degrees in the graph. Gutman and Trinajstić introduced the first Zagreb index in 1972, and originated from the study of chemical graph theory, where each vertex in the graph has a degree of less than or equal to four. The first Zagreb index is one of the most important topological indexes in the chart and has been studied in depth for many years.
If the graph has a loop containing all the vertices in the graph, it is called a Hamiltonian graph. The Hamiltonian problem in graph theory is to find the features of the Hamiltonian graph. Mathematically, it is to find a sufficient and required condition for the Hamiltonian diagram. The Hamilton problem is a major unsolved problem in graph theory. When investigating the Hamiltonian problem, researchers usually focus on finding sufficient conditions for Hamiltonian diagrams.
Recently, Professor Rao Li from the University of South Carolina Aiken University proposed new conditions based on the first Zagreb index of the Hamiltonian graph. The study has been published in the peer-reviewed journal Mathematics. During the research period, Professor Li used the famous chvátal-erdös theorem in Hamilton’s graph theory, an observation on the graph, and two inequalities determined by Shisha and Mond in 1967.
It is generally believed that it is difficult to find enclosed mathematical expressions for the first Zagreb index of the graph. Researchers often focus on getting the boundaries of the first Zagreb index. Professor Li realized that ideas and techniques developed in terms of sufficient conditions for obtaining Hamiltonian diagrams could be used to establish a new upper limit for the first Zagreb index. After careful analysis, Professor Li finally provided two new achievable upper limits for the first Zagreb index in the same paper.
“It is interesting that we were able to use inequality in mathematical analysis to find new sufficient conditions involving the first Zagreb index of the Hamiltonian graph, as well as a new upper limit for the first Zagreb index of the graph. This study shows a new application of the first Zagreb index and enriches the research on Hamiltonian graph theory and the first Zagreb index of the graph,” said Professor Li.
Journal Reference
Li, R. “The first Zagreb index and the Hamiltonian properties of some graphs.” Mathematics, 2024. Doi:
