Study On Extremal Problems Of Two Classes Of Graph Parameters

The topological indices are numerical quantities closely related to chemical structure.They are widely used in mathematical chemistry,especially in the study of QSPR/QSAR.Therefore,they have very important research value and significance.Topological indices,including Wiener index,Szeged index,edge Szeged index,have been extensively and deeply studied.The edge Szeged index is based on related properties of Szeged index,which was proposed by famous mathematician Gutman in 2008.Later,a lot of scholars focused on the edge Szeged index and have obtained fruitful results which enrich related theories.The general position problem is based on the classical Dudeney's no-three-in-line problem,which was proposed by the academician of international academy of mathematical and chemistry Sandi in 2018.At present,many scholars are very concerned about general position problem.In the thesis,based on the previous results,we discuss the edge Szeged index and general position problem of graphs.Firstly,we obtain the sharp bounds of edge Szeged index regarding bicyclic graphs.Meanwhile,we completely characterize extremal graphs that meet the bounds.Secondly,the sharp bounds of gp-number for cactus are determined,and the structure of the extremal graphs that attain the bounds are also characterized.Finally,the gp-number is gotten for wheel graph.The paper can be divided into the following four chapters.Chapter 1: Firstly,we give the background and significance.Secondly,we discuss the status of study at home and abroad.Then,we give relevant concepts and definitions,In the end,we summarize the structure of the thesis.Chapter 2: Firstly,for edge Szeged index,we list its present conclusions on unicycle graphs and obtain its some basic properties among bicyclic graphs.Secondly,we deduce the sharp bounds of edge Szeged index in bicyclic graphs.In addition,we characterize these graphs that meet the bounds.Chapter 3: Firstly,we deduce the some lemmas of general position problem for cycles and pendant trees of cactus graphs.Secondly,we determine the sharp bounds of gp-number for cacti,and also characterize the structure of the extremal graphs that attain these bounds.Finally,the gp-number of wheel graphs is gotten.Chapter 4: We summarize the main content of the thesis,and set future research directions and learning goals.