On The Extremal Problems For The General Randi(?) Index

It is well known that graph theory has a close relation with chemistry. Tn fact, it is natural to use a graph or a tree to represent a molecule when the only property considered is the existence or not of a chemical bond. Recently, people have tried to get new medicines with desired physico-chemical properties. So, combinatorial cliemistrv has played a powerful role in drug design and molecular recognition.A topological index is a map from the set of chemical graphs to the set of (positive) real numbers. Therefore, a topological index is a numeric quantity that is mathematically derived in a direct and unambiguous manner from the structural graph of a molecule. The advantage of topological indices is in that they may be used directly as simple numerical descriptors in a comparison with physical, chemical, or biological parameters of molecules in quantitative structure-property relationships (QSPR) and in quantitative structure-activity relationships (QSAR).The design of new molecules with desired properties can be guided by finding the structures of molecular graphs with some given values of a certain topological index. A natural problem is to determine the range of a certain index and the structure of graphs with the extremal index. These extremal problems become more and more interesting and important in this field. The solutions of these questions will help to determine the range for computer searching. The main task for graph theorists is to set up the molecular database, and provide the data base for chemists to wisely synthesize new drugs.In 1975, Milan Randic proposed a chemical index, the connectivity index (also called the Randic index). As one of the many graph-theoretical parameters derived from the graph underlying some molecule, Randic index has a good correlation with several physicochemical properties of alkanes: boiling points, chro-...