Mathematics and chemistry make excellent partners.
Dennis H. Rouvray
Editional Foreword
J. Math. Chem. 1 (1987)
Mathematical chemistry has a long history extending back to the times of Russian polymath Mihail Vasiljevič Lomonosov (1711-1765), when he attempted in the mid 18th century to mathematize chemistry [1]. A part of mathematical chemistry that we call chemical graph theory [2] has also a distinguished past that extends to the second half of the 19 th century when Arthur Cayley (1821-1895) was enumerating alkane isomers [3] and James Joseph Sylvester (1814-1897) introduced the terms algebraic chemistry and graph [4,5]. Alexander Crum Brown (1838-1922), who was trained in both chemistry and mathematics, was probably the first chemist who did research in mathematical chemistry [6,7].
The term algebraic chemistry has in due course been replaced by the more general term mathematical chemistry, but a better term than graph has never been found. The seminal role of Cayley and Sylvester in the early development of mathematical chemistry in general and chemical graph theory in particular has been expertly reviewed by Dennis H. Rouvray [8]. It is important to point out why mathematical chemistry is relevant to chemistry. We could not do that better than Jerome Karle, Nobel Prize Laureate 1985, who wrote [9]: Mathematical chemistry provides the framework and broad foundation on which chemical science proceeds.
Mathematical chemistry and chemical graph theory were developing sluggishly with only a few leaps, such as Pólya's work on combinatorial enumeration [10], until the 1970s. Then there suddenly appeared several research groups, located world-wide, who started to speedily develop chemical graph theory. One of the directions in which this vigorous revival was moving was the introduction of a number of novel graph-theoretical matrices.
Matrices are the backbone of chemical graph theory. Classical graph-theoretical matrices are the (vertex-) adjacenc ymatrix, the (vertex-edge) incidence matrix and the (vertex-) distance matrix [11-19]. Historically, incidence matrices appear to have been the first to be used [20]. However, the most important graph-theoretical matrix is the vertex-adjacency matrix as is well-illustrated by Cvetković, Doob and Sachs in their monograph Spectra of Graphs – Theory and Applications [19], the first edition of which appeared in 1982. An important source for the distance matrix is the monograph Distance in Graphs by Buckley and Harary [18].
In last 25 years perhaps more than one hundred novel graph-theoretical matrices have been introduced. Among the literature sources reporting some of these matrices and their uses are monographs: Handbook of Molecular Descriptors by Todeschini and Consonni [21] , Topological Indices and Related Descriptors in QSAR and QSPR, edited by Devillers and Balaban [22] and MolecularTopology by Diudea, Gutman and Lorentz [23], and review articles: Molecular graph matrices and derived structural descriptors by O. Ivanciuc, T. Ivanciuc and Diudea [24] and Eingenvalues as molecular descriptors by Randić , Vračko and Novič [24a].
We present 130 graph-theoretical matrices in the encyclopaedic manner, classified into five groups: adjacency matrices and related matrices, incidence matrices, distance matrices and related matrices, special matrices and graphical matrices. The motivation for preparing this monograph comes from the fact that among the matrices presented several are novel, several are known only to a few and the properties and potential usefulness of many graph-theoretical matrices in chemistry are yet to be investigated.
Most of the graph-theoretical matrices that we present here have been used as sources of molecular descriptors usually referred to as topological indices — the term topological index was introduced 35 years ago by Hosoya [25] — which have found considerable application in structure-property-activity modeling [2,21-23,26-28], usually abbreviated as QSPR (quantitative structure-property relationship) [29] and QSAR (quantitative structure-activity relationship) [30]. Graph-theoretical matrices have also been used however for many other purposes in chemistry [e.g., 31-39].
Hopefully, this monograph will stimulate some readers to undertake research in this fruitful and rewarding area of chemical graph theory and introduce new kinds of graph-theoretical matrices that may find use in chemistry.
Finally, we wish to point out that this book is an outcome of the long-standing Croatian-Slovenian joint research collaboration in computational and mathematical chemistry.
The authors would also like to thank G.W.A. Milne, former Editor-in-Chief of the Journal of Chemical Information and Computer Sciences, for his editorial assistance with this book.