The Vertex Gutman Index and Gutman Index of the Union of Two Cycles

The Wiener index is one of the most classic and widely used indicators in topology. It reflects the average distance of any node pair in the graph. It not only makes the boundaries of given graphs clearer but also continuously generates topological indices that are more suitable for new fields, such as the Gutman index. The Wiener index and Gutman index are two important topological indices, which are commonly used to describe the characteristics of molecular structure. They are closely related to the physical and chemical properties of molecular compounds. And they are widely used to predict the physical and chemical properties and biological activity of molecular compounds. In this paper, we study the vertex Gutman index and Gutman index and describe the structural characteristics of all cases of two simple cycles intersecting. We comprehensively analyze the Gutman index and vertex Gutman index in these cases in detail by means of classification discussion and analogical reasoning and characterize their maximum and minimum accordingly.