Research On Topological Indices Of Graphs

Topological index is a numerical value related to molecular structure.The first molecular topological index was proposed by Wiener in 1947,now known as Wiener index.In the past few decades,topological index has developed rapidly and made great achievements.Moreover,many scholars cross study topological indexes with other disciplines,which has played a very important role in the development of other disciplines.For example,Wiener index can be used to predict the boiling point of organic compounds in chemistry.In 2016,as a generalization of Wiener index,the Steiner Wiener index was introduced by Li,Mao and Gutman,and many research achievements have been made now.There are five chapters in this thesis,which are organized as below.In Chapter 1,we first introduce the background of topological index and the research progress of related problems involved in this thesis.In Chapter 2,we study the Steiner Wiener index of graphs.The Wiener index W(G)of a connected graph G is defined as W(G)=∑{u,v}?V(G)dG(u,v),where dG(u,v)is the distance between the vertices u and v in G.For S?V(G),the Steiner distance d(S)of the vertices of S,introduced by Chartrand,Oellermann,Tian and Zou in 1989,is the minimum size of a connected subgraph of G whose vertex set contains S.For an integer k≥1,the Steiner k-Wiener index of G,denoted by SWk(G),is ∑S?V(G),|S|=kd(S).Clearly,SW2(G)=W(G)for a connected graph G.In 2016,Li,Mao,and Gutman proved that for any tree T,Using Vandermonde’s convolution formula,we reformulate it as for any tree T of order n.Thereby,we determine the minimum and the maximum Steiner k-Wiener index of trees with given bipartition.This extends the results on Wiener index of trees with given bipartition due to Du.We give sharp bounds of SWk(G)of unicyclic graphs with fixed k=n-1,k=n-2 and k=3 and we characterizes the extremal graphs.In Chapter 3,we study the arithmetic-geometric index of bicyclic graphs.In the mathematical and chemical literature,many vertex-degree-based graph invariants have been introduced and extensively studied.They may be uniformly expressed as where F(x,y)is a function with the property F(x,y)=F(y,x).In 2015,Shegehall and Kanabur defined a graph invariant,called the arithmetic-geometric index:They calculated the value of the AG index for some special graphs.In 2018,Milovanovic,Matejic,and Milovanovic obtained several upper bounds for the AG index of some graphs.In 2021,Vujosevic,Popivoda,Vukicevic,Furtula and Skrekovski deals with some lower and upper bounds of AG index for trees,connected graphs and connected bipartite graphs.Besides,in the same paper,the authors gives the characterization of chemical trees with maximize the value of AG index,and gives some relations between the AG index and GA index.In 2021,Vukicevic,Vujosevic and Popivoda determined the maximum AG(G)index and the minimum AG(G)index of connected unicyclic graphs with n vertices,and n>3.Furthermore they put forward a conjecture:If G is a connected bicyclic graph of order n,then The lower bound is attained by the next two families of graphs:family Cn+ of graphs obtained by adding one chord to a cycle Cn and family Cks of graphs consisting of cycles Ck and Cs connected by a cut-edge,for each k,s,such that s+k=n.The upper bound is attained by the graph obtained from C4+ attaching n-4 pendant vertices on the one of its vertices with degree 3.We confirm the validity of the conjecture.In Chapter 4,we study the geometric-arithmetic index of bicyclic graphs.In 2009,the geometric-arithmetic index is a vertex-degree-based graph invariants introduced by Vukicevic and Furtula as a modification of the well known Randic index,and it is defined as In 2011,Das,Gutman and Furtula give some lower and upper bounds on GA index in terms of the order n,the size m,the minimum degree δ,maximum △ and other topological index.In 2017,Aouchiche and Hansen presented some bounds on GA index in terms of the order n,the chromatic number χ,the minimum degree δ,maximum △ and average degree d.In 2011,Du,Zhou and Trinajstic obtained the upper bounds for the GA index of connected bicyclic graphs with order n.We prove that:If G is a connected bicyclic graph of order n,then The upper bound is attained by the next two families of graphs:family Cn+ of graphs obtained by adding one chord to a cycle Cn and family Cks of graphs consisting of cycles Ck and Cs connected by a cut-edge,for each k,s,such that s+k=n.The lower bound is attained by the graph obtained from C4+ attaching n-4 pendant vertices on the one of its vertices with degree 3.In Chapter 5,we study the Sombor index of graphs.The Sombor index SO(G)of a connected graph G,introduced by Gutman in 2021,is defined as where d(vi)and d(vj)are the degree of the vertices vi and vj in G.Furthermore Gutman obtained:If T is a tree of order n,then with the left-side of equality if and only if T(?)Pn,and with the right-side of equality if and only if T(?)K1,n-1.In 2021,Cruz,Rada prove that:If G is a connected unicyclic graph of order n,then SO(Gn)=2(?)≤SO(G)≤SO(U(n,n-3,0,0)).In 2021,Chen,Li and Wang prove that:(1)If T is a tree with given diameter d,then with the equality if and only if T(?)Tn,d1.(2)If T is a tree with given(p,q)-bipartition,then with the equality if and only if T(?)S(p,q).We prove the above results by different ways.