On The Minimal Edge-szeged Index Of Unicyclic Graphs With A Given Diameter

Given a connected graph G,the edge-Szeged index Sze(G)is defined as Sze(G)=?e=uv?Emu(e)mv(e),where mu(e)and mv(e)are,respectively,the number of edges of G lying closer to vertex u than to vertex v and the number of edges of G lying closer to vertex v than to vertex u.In this paper,some extremal problems on the edge-Szeged index of unicyclic graphs are considered.All the n-vertex unicyclic graphs with a given diameter having the minimum edge-Szeged index are identified.Using a unified approach we identify the n-vertex unicyclic graphs with the minimum,second minimum,third minimum and the fourth minimum edge-Szeged indices.The concrete content is in the following:· In Chapter 1,we introduce the background and significance of the research,in-cluding the development of a representative at home and abroad regarding this aspect.Based on this research background and profound discussion,by using deep-going analysis,it fully shows the main work's necessity and innovation.· In Chapter 2,we give some necessary definitions and lemmas.· In Chapter 3,the sharp lower bounds on the edge-Szeged index and the ex-tremal graphs in the class of all n-vertex connected unicyclic graphs with a given diameter d are characterized.· In Chapter 4,the minimum,second minimum,third minimum and the fourth minimum edge-Szeged indices and the extremal graphs of n-vertex unicyclic graphs are all determined.· In Chapter 5,we summarize the main results in this paper and give some prospects for further research in the future.