The Extremum Problem Of Exponentials In Geometric Arithmetic Of Graphs

Graph theory is one of the important branches of mathematics.It has gotten great attention in recent decades.It has a wide variety of applications in many fields.Nowadays,the combination of graph theory with computer science,chemistry and other fields has produced many interdisciplinary subjects,such as chemical graph theory,which plays a crucial role in the field of chemistry.Geometric arithmetic index is a classical topological index in chemical graph theory,which has extensive applications in QSPR and QSAR of compounds.We mainly study the extremum problem of geometric arithmetic index in this paper.In the second chapter,we study the inequality relationships between geometric arithmetic index,Zagreb index,harmonic index,sum connectivity index and NK index.We use algebraic formulas containing these indices to give some new upper and lower bounds of the geometric arithmetic index.At the same time,with the help of previous research results on these topological indices,We give some new upper and lower bounds of the geometric arithmetic index and the corresponding extremal graphs.In the third chapter,we give the upper bound of the geometric arithmetic index of the pentacyclic graphs and the structure of the corresponding extremal graphs.In the fourth chapter,we find that among the graphs with given chromatic number or clique number,the graph with the largest geometric arithmetic index can only be a Turán graph.