The Extremal Problem Of Connectivity Index In Trees With K Pendant Vertices

Some topologic indices of a molecular structure can reflect its physico-chemical properties.Different topologic index reflects different properties. They make great contributions to the study of QSPR (quantitative structure-property relations) and QSAR (quantitative structure-activity relations). To research into the relationship between the structure of a molecule and its physico-chemical properties by mathematical methods, researchers put forward to some topological indices of molecular structure which relate to its physico-chemical properties. The most well-known are Wiener index, Hosoya index and Randic index.A molecular structure can be regard as a graph G. Randi(?) index is a graph invariant defined aswhere d_G(u) denotes the degree of a vertex u in the graph G, and the summation goes over all pairs of adjacent vertices of G.In 1975, in studying the extent of branching of the carbon-atom skeleton of saturated hydrocarbons. Milan Randi(?) proposed an important molecular topological index-branching index. The quantity R is well correlated with a variety of physico-chemical properties of molecule, and makes great contributions to the stduy of QSPR and QSAR. Randi(?) noted that R is well correlated with a variety of physico-chemical properties of alcanes. Randi(?) found when C_nH_2n+2 is decreasing by R(G), their extents of branching is increasing. The Randi(?) index can measure the extent of branching of the carbon-atom. This influences the physico-chemical properties of molecule.From a mathematical point of view, the first question to be asked is which are the extremal R-values in classes of graphs, and which are the graphs from these classes with extremal R. Finding the answers is not easy, and sometimes the extremal graphs have quite unusual and interesting structures. Many chemists and mathematicians are doing re- searches in finding the answers,because they noticed that there is a good correlation between the Randi(?) index R and several physico-chemical properties of a molecule. In subsequent years countless applications of R were reported,most of them concerned with medicinal and pharmacological issues.In this thesis, we study the extremal problem of index in trees with k pendant vertices. The trees of order n with k pendant vertices which have maximum Randi(?) index in the case n≥3k-2 were characterized in [11]. We study the case n<3k-2.We use some proper transformations to study the problem of extremal values of the Randi(?) index of trees with n order and k pendant vertices(n<3k-2).We can find some extremal trees with n order and k pendant vertices (n<3k-2) quickly and easily by making use of the properties we find.