Wiener-type Indices Of Hamiltonian Graphs

In chemical theory, the topological index can be used to understand the physical and chemical properties of compound. Di?erent indices re?ect di?erent properties of the molecules. The study of molecular topological index and invariant of molecular graphs is one of the research ?elds of chemical graph. Every vertex of simple undirected graph G =(V, E) represents an atom, every edge represents chemical bonds between atoms,this graph is known as a molecular graph. In a molecular graph, we put the vertex number and edge number as a stable invariant in the molecules. In practically, di?erent values describe the di?erent measurable molecular chemical and physical properties, so to connect the topological properties with measurable physical and chemical properties of the molecule, it is necessary to introduce some numeric quantity, which are associated with certain properties of molecular graph. Molecular topological indices have important application in physics, chemistry, biology and many other subjects. The Wiener index is one of the most intensively studied topological indices. It is not only an early topological index which correlates well with many physico-chemical properties of organic compounds but also a subject that has been studied by many mathematicians and chemists.The Wiener index [25] is the sum of distances between all unordered pairs of vertices of a connected graph, i.e. W(G) =∑{u,v}?VdG(u, v) =12∑(u,v)∈V ×VdG(u, v),where dG(u, v) is the distance between u and v in G. The hyper-Wiener index of a connected graph is de?ned as W W(G) =12W(G) +12∑{u,v}?Vd2G(u, v) =12W(G) +14∑(u,v)∈V ×Vd2G(u, v), where d2G(u, v) = dG(u, v)2. In [12, 13] the multiplicative version of the Wiener index(also called π-index) was conceived by Gutman et al.: π(G) =Π{u,v}?V(G)dG(u, v). Let Hnbe the set of Hamiltonian graphs of order n. In this paper,we characterize the elements in Hnwith the smallest and the i-th smallest(1 ≤ i ≤ n- 2).Wiener, hyper-Wiener and π- indices. Moreover, we determine the graphs in Hnwith the greatest, the second and the third greatest Wiener, hyper-Wiener and π- indices, and index calculate formulae are also given.