Research On Extremal Values And Rankings Of The Gutman Index Of Some Trees

Topological index of graph plays an important role in molecular characterization and the establishment of relationship of molecular structure and characteristic.Topological index of graph,widely used to predict the physicohemical property and bicoactivity of compounds,is a topological index closely related to physicohemical property of compounds,so it is universally applied in the field of chemistry.The Wiener index is a very classic topological index that is related to the distance between the vertices of a graph.The weighting of it can derive a lot of topological indexes.The Gutman index is a graph topology index derived from the Wiener index,which is related to the distance and degree of the vertex.Its proposal extends the field of research on the topological index related to distance and degree of graph.In this paper,the Gutman indexes of several typical trees are studied.The main work and some important results are as follows:(1)We studied the vertex-Gutman index of extended Double Stars and the Gutman index of the family of extend Double Stars,and we also gave their general calculation formulas.We obtained the extremal values and changes of vertex-Gutman index of extended Double Stars.We also provided the extremal values and rankings of Gutman index of the family of extended Double Stars,and then we characterized the corresponding extremal graphs.(2)We studied the vertex-Gutman index of advanced k-Caterpillars and the Gutman index of the family of advanced k-Caterpillar Trees and gave their general calculation formulas.According to structural characteristics of advanced k-Caterpillar Trees,the extremal values and rankings of which are studied,and then corresponding extremal vertices and extremal graphs are characterized.(3)Based on the analysis of the structural characteristics of Newkome-type Trees,the general calculation formulas of vertex-Gutman index of Newkome-type Trees and Gutman index of the family of Newkome-type Trees are given by means of overall segmentation and step calculation.