Some Extremal Problems Of Wiener Indices Trees

For a connected graph G,the Wiener index of G is defined to be the sum of the distances between all pairs of vertices in G.The Wiener index was first proposed by Harold Wiener as an aid to determining the boiling points of alkanes,which is also called a molecular structure descriptor or molecular topological index.Molecular structure descriptors or molecular topological indices are nowadays extensively used in theoretical chemistry for the design of so-called quantitative structure-property relation(QSPR)and quantitative structure-activity relation(QSAR).In addition, the Wiener index also has many applications in communication,facility location, cryptology,etc.The Wiener index has been extensively studied since the middle of the 1970s.In chemical applications and mathematics,it is of great important to identify the graphs with extremal Wiener indices.And many results have been obtained.In this paper,we consider the trees with some given parameters(order,diam-eter,maximum degree,independence number and matching number)and extremal (minimum,maximum,the second minimum,the second maximum,etc.)Wiener indices,and give the following results:1.determine the tree with the minimum Wiener index among all the trees with diameter d and order n;2.determine the trees with the minimum and the maximum Wiener indices among all the trees with diameter≥d and order n;3.determine the trees with the minimum and the maximum Wiener indices among all the caterpillar trees with diameter d and order n;4.determine the tree with the maximum Wiener index among all the trees with maximum degreeΔand order n;5.determine the trees with the second and the third maximum Wiener indices among all the trees of order n whose vertices have degree 1 orΔ;6.determine the trees with the minimum and the second minimum Wiener indices among all the trees with independence numberα(or matching numberβ) and order n.