| Let G=(V(G),E(G))be a simple graph,dG(v)be the degree of the vertex v in G,and dG(u,v)be the distance between the vertices u and v in G.As a representative of distance-based graphical indices,the Wiener index has several generalized forms,the most famous of which are the Gutman index and the degree distance.Various graphical indices have been proposed over the decades,and many of them have similar or even identical extremal results and proof techniques.So,general graphical indices have emerged for a unified discussion of a large class of graphical indices.Based on the Gutman index and degree distance,the general Gutman index(Gutα,β(G)=∑u,v∈V(G)(dG(u)dG(v))αdG(u,v)β,α,β∈R)and general degree distance(DDα,β(G)=∑u,v∈V(G)(dG(u)α+dG(v)α)dG(u,v)β,α,β∈R)were proposed,respectively.In addition,a more general graphical index,the vertex degree function index(Hf(G)=∑v∈V(G)f(dG(v)),where f is a real-valued function defined on the vertices of a graph G),was also proposed.In this paper,we focus on the bounds of the general Gutman index,the general degree distance,the vertex degree function index,the modified Wiener index(Wβ(G)=∑u,v∈V(G)dG(u,v)β,β∈R),and the extremal values of the degree distance for trees with given segment sequence.The main results of this paper are as follows.1.A tree is said to be star-like if only one vertex has a degree greater than 2.Das et al.gave bounds of the general Gutman index for star-like trees under certain values of α and β.In this paper,we give and analyze three operations on graphs,by which we characterize the case that the general Gutman index achieves extremal values for trees with given order under some values of α and β.In addition,we give bounds of the general Gutman index for trees with given diameter.2.Das et al.gave the bounds of the degree distance,the difference between the Gutman index and the degree distance by Wiener index,the first Zagreb index,the second Zagreb index,etc.In this paper,we give the bound of the general degree distance for trees by modified Wiener index.For general graphs,bounds on the difference between the general degree distance and the general Gutman index are given by other general graphical indices.In addition,we give bounds of the general degree distance for trees with given diameter.3.Hu et al.gave bounds of the vertex degree function index for graphs with given order and number of edges.In this paper,we give bounds of the vertex degree function index for connected graphs with given order,number of edges,maximum degree and minimum degree,and we also give construction methods for the extremal graphs.Andriantiana et al.discussed the class of graphs that achieve the maximum of the Wiener index for trees with given segment sequence and characterized the extremal trees.As a generalization of the Wiener index,we discuss the class of trees with given segment sequence that maximize the degree distance,and furthermore we characterize the extremal trees that achieve the maximum.In addition,we give a simple estimate of the modified Wiener index of graphs. |
