Bicyclic Graphs With Maximum General Randic Index

In order to study the extent of branching of the carbon-atom skeleton of saturatedhydrocarbons in 1975 the famous chemist Milan Randi(?) proposed an important topologicalindex–branching index, defined as where d(u) denotes the degreeof a vertex u. It is also called connectivity index or Randi(?) index. Randi(?) index is closelycorrelated with a variety of physico-chemical properties of molecular(such as boiling point,surface etc.) and pharmacologic properties. The general Randi(?) index was proposedby B. Bollobás, P. Erdo(o|¨)s and D. Ami(?) et al., independently, in 1998. It is defined as whereαis any real number. Note that the Randi(?) index isa special case of the general Randi(?) index. The general Randi(?) index has attracted thewidely attention of both theoretical chemists and mathematicians, and a lot of resultshave been obtained. The main work are concerning the problems of determining thebounds for the general Randi(?) index of a given class of graphs as well as related problemsof finding the graphs with maximum or minimum general Randi(?) index. A graph withn vertices and n + 1 edges is called a bicyclic graph. In this thesis, we characterize thebicyclic graphs with maximum general Randi(?) index forα≥1.