On The Harmonic Index And Zagreb Eccentricity Indices

The Randic index, harmonic index and two Zagreb eccentricity indices are the most important indices in the chemical and mathematical fields. The Randic index is the widely applied molecular descriptor and mainly used in the study of structure-property and structure-activity relationships. The harmonic index is a variant of Randic index, the harmonic index H(G) of a graph G is defined as the sum of the weights2/(a(u)|d(v))of all edges uv of G, where d(u) denotes the degree of a vertex u in G. some study show it gives somewhat better correlations with physical and chemical properties than Randic index. The first Zagreb index and second Zagreb index are graph-based molecular structure descriptors, they are defined as:M1(G)=?d(u)2and M2(G)=?d(u)d(v), they reflect the branching extent of molecular skeleton and correlate well with energy of molecular. The Zagreb eccentricity indices are the variant of the Zagreb indices, they are defined as:ZEX1(G)=??G(u)2, ZE2(G)=??G(u)?G(V), where eG(u) denotes the eccentricity of u, it is the maximum distance dG(u,v) between any two vertices u and v.The full text is divided into five parts.In the first part, we introduce the research background, current situation and main content of the paper.In the second part, we introduce some basic concepts, symbol marks and related theorems in the graph, and some properties and theorems of the tree and related graph. we also explain some operations of Cartesian product.In the third part, we study the application of harmonic index in the special graphs.In this part, we present a brief proof of the conclusion that the path Pn has the maximum harmonic index in the tree graph, the new method is different from the other. The previous method to prove it by the mathematical induction, in this paper, we transform the edges in the simple connected graph to get the conclusion. When comparing, this one is more concise than the other and easier to generalize to other situations, such as unicyclic graphs and bicyclic graphs and so on. According to this idea, we get the extremum of unicyclic graphs and bicyclic graphs, and characterize the corresponding extremal graphs. Then we give a further discussion about the propertyof the graphs with the maximum harmonic index, and show that the regular or almostregular graphs have the maximum harmonic index.In forth paper, we concentrate on the application of the first and second Zagrebeccentricity indices in the Cartesian product.In this part, we get the expression of two Zagreb eccentricity indices in the Carte-sian product, and also present the expression of the Zagreb eccentricity indices in arbi-trary Cntube and Cntorus.In the fifth part, we get the conclusion and summary of this paper and outlook ofthe future.