Mathematical Chemistry Monographs,
University of Kragujevac, Kragujevac 2006,
330 pp, hard cover, $120.00
ISBN 86-81829-67-

Nenad Trinajstić

English

Hrčak - Portal of scientific journals of Croatia

BOOK REVIEWXueliang Li, Ivan GutmanMathematical Aspects of Randi}-typeMolecular Structure DescriptorsMathematical Chemistry Monographs,University of Kragujevac, Kragujevac 2006,330 pp, hard cover, $120.00ISBN 86-81829-67-XOne of the contemporary leaders in mathematical chemi-stry Professor Ivan Gutman (Faculty of Science, Universityof Kragujevac, Kragujevac, Serbia) produced, in collabo-ration with the Chinese mathematician Professor XueliangLi (Center for Combinatorics, Nankai University, Tianjin,China), the first book in the series Mathematical Chemi-stry Monographs. In this book, the authors consider theoriginal Randi} connectivity descriptor that Professor Mi-lan Randi} proposed in 1975 and Randi}-type descrip-tors. Immediately after being published, Randi}’s famouspaper entitled "On Characterization of Molecular Branch-ing" (J. Am. Chem. Soc. 97 (1975) 6609–6615) attractedattention of mathematical chemistry community, but alsoof computational chemists, environmental chemists, peopledoing research in drug design and many other chemistsinterested in encoding molecular structure by a singlenumber, and later even by mathematicians such as BélaBollobás and Paul Erdös. In due course, this paper be-came the most cited mathematical chemistry paper ofmodern times and was listed as the 95th paper in thecompilation of 125 most cited papers that appeared inthe Journal of the American Chemical Society in the last125 years. While initially chemists used this index most-ly in its original form, mathematicians later investigatedits mathematical properties. The present book is aboutmathematical aspects of the Randi} descriptor and itsgeneralizations.The book consists of six chapters and bibliography.The first chapter, entitled Introduction (pp. 1–10),contains a foreword by Milan Randi}, introductorynotes, the outline of the book and basic terminology andnotation.The second chapter, entitled The Randi} index (pp.11–107) considers the Randi} index in the originalformulation:R(G) = [ n ]d u du E( ) ( ) /( )−∈∑ 1 2n Gwhere R(G) is the Randi} index of a graph G, u,n arevertices of G, d(u) and d(n) are degrees of vertices u andv, E denotes the number of edges in G and the summa-tion goes over all pairs of adjacent vertices in G. Li andGutman adopted the symbol R for the Randi} index in-stead of the more commonly used c. The chapter is di-vided into several sections, each dealing with a class ofgraphs. The considered graphs are general graphs, trees,unicyclic, bicyclic and tricyclic graphs, chemical graphsand chemical trees. Various mathematical properties,such as minimum and maximum values of the Randi}index for some graphs, bounds for trees of a given order,bounds of some chemical graphs, etc., are reported.The third chapter, entitled The General Randi} In-dex (pp. 109–273), considers the generalized form of theRandi} index:Ra(G) = [ n ]ad u du E( ) ( )( )n ∈∑Gwhere a is a real number. If we put the value a = –1/2 inthe above formula, then we get the original Randi} index.On the other hand, if we put a = 1, we get the secondZagreb index (see I. Gutman and N. Trinajsti}, "GraphTheory and Molecular Orbitals. III. Total p-ElectronEnergy of Alternant Hydrocarbons," Chem. Phys. Lett. 17(1972) 535–538; I. Gutman, B. Ru{~i}, N. Trinajsti} andC. F. Wilcox, Jr., "Graph Theory and Molecular Orbitals.XII. Acyclic Polyenes," J. Chem. Phys. 62 (1975) 3399–3405; for some reason these references are not cited inthe book). Graphs considered are general graphs, uni-cyclic graphs, trees and chemical trees. And again, manymathematical properties, such as the trees with mini-mum and maximum values of the general Randi} index,chemical trees with minimum and maximum values ofthe R–1 index, etc., are discussed.The fourth chapter, entitled The Zeroth-orderRandi} Index (pp. 275–309), reports on the properties ofthe Randi} index of the zeroth-order:0R(G) = [ ]d uu( ) /( )−∑ 1 2Gand the general Randi} index of the zeroth-order:0Ra(G) = [ ]d uu( )( )aG∑CROATICA CHEMICA ACTACCACAA 79 (3) A31–A32 (2006)where the summation in both above formulae goes overall vertices in G. The general zeroth-order Randi} indexfor a = 2 gives the first Zagreb index. In this chapter, theauthors discuss general graphs with maximum 0R and 0Raindices of trees, unicyclic graphs with the first three ex-tremal 0R and 0Ra indices of chemical graphs and che-mical trees.The fifth chapter, entitled Higher-order Randi} In-dices (pp. 311–317), offers a short discussion about them-th order Randi} index:mR(G) = [ ]d d di i imi i im1 2 11 21 2 1... /...+−+∑where the summation is over all paths i1i2...im+1 of lengthm, contained in G. The mR is considered for generalgraphs.In the last chapter, Conclusion (pp. 319–322), Liand Gutman state that they have examined various ex-tremal problems related to the Randi} index and its va-rious generalizations. They were particularly interested intwo problems: to find the lower and upper bounds for theRandi} indices (R, Ra,0R, 0Ra,mR, m>1) and to charac-terize graphs for which the Randi} indices assume the ex-tremal (minimum or maximum) values. In this chapter,the authors also pose a few open problems that they findmost interesting and challenging from the mathematicalpoint of view.The book ends with Bibliography (pp. 323–330) con-taining 105 references given in alphabetical order. TheZagreb group papers, related to the subject of the book,are cited 5 times. It appears that the weakest point of thisotherwise very carefully prepared book is the bibliography,because a number of relevant papers are not mentioned.It is also announced that three more books in the se-ries Mathematical Chemistry Monographs are already inpress. Information about these three forthcoming books canbe found at the address http://www.pmf.kg.ac.yu/match/.We are looking forward to other books in the series withgreat anticipation because the first beautifully designedbook is so rich in interesting and stimulating results.Nenad Trinajsti}A32 BOOK REVIEWCroat. Chem. Acta 79 (3) A31-A32 (2006)

Xueliang Li, Ivan Gutman: Mathematical Aspects of Randić-type Molecular Structure Descriptors

Abstract

Similar works

Full text

Available Versions

Hrčak - Portal of scientific journals of Croatia

HRČAK - Portal of Croatian Scientific and Professional Journals