# Graph-Theoretical Matrices in Chemistry

Mathematical chemistry is an exponentially expanding field, not in the least because of the enormous interest in nanoscience these days. Very early it was recognized that graphs were an excellent tool to describe molecular structures. To analyse these graphs, linear algebra and matrices play an important role. No wonder that a first version of this book appeared already a while ago in 2007 (at the U. of Kragujevac). Since this book was sold out, and novel graph-theoretical matrices were introduced since then, the authors decided to publish this enlarged edition.

The best known, and most important matrix in this context is the (vertex) adjacency matrix, but also the (vertex) distance and the (vertex-edge) incidence matrix are probably known too. And these show up in many variants. During the last decades some 100 new matrices were introduced that turned all to be important for several applications. Some 170 different matrices are collected in this book. They are grouped in five different chapters entitled 'The adjacency matrix and related matrices', 'Incidence matrices', 'The distance matrix and related matrices', 'Special matrices', and 'Graphical matrices'.

Each chapter is an encyclopedic enumeration of the matrices with their definition, which is usually followed by a very detailed list of references where they were introduced and for what purpose they are used. Almost all the matrices that were defined are numerically illustrated for a few simple graphs that are used throughout the book, which is a great help to properly understand the definitions. However how to compute them efficiently or a more elaborate illustration of their use in further applications is not included, only references are provided. For example, one of the 'Special matrices' is the Wiener matrix, a symmetric matrix defined for an acyclic graph. Suppose the edge connecting vertices *i* and *j* is removed. Then count the number of vertices that are still connected to *i*, including *i* itself and the number of vertices connected to *j* including *j*. The product of these numbers is the entry (*i,j*) of the matrix. Its diagonal entries are zero. The Wiener index is an important topological parameter that is obtained by summing the elements in the upper (or lower) triangular part of the Wiener matrix. This Wiener index shows up at other places too in this book, but when you want to look up 'Wiener index' in the index, it is not there. All the matrices are there, yes, but not all the applications for which they can be used. There is no explanation what the Wiener index really means (e.g. centrality of a node in a network). Anyway, the Wiener matrix was not really needed in the first place. The Wiener index could as well be defined as a sum of products without needing the matrix. There are other, equivalent definitions of the Wiener index possible. For example in terms of a distance matrix whose elements are the lengths of the shortest path between any pair of vertices in the graph. So the Wiener index is also mentioned in connection with a distance matrix, but it is not said how it relates exactly to the matrix. Only a long list of references is given. Similar observations can be made in relation to other concepts and matrices as well. The example of the Wiener index is just to illustrate what the book does and what it does not contain. To look up something via the index, you need to know the name of the relevant matrix. A preliminary online version (2009) of this book is also available at www.sicmm.org/~FAMNIT-knjiga. This site has a search button, but that doesn't seem to work.

I believe the titles of the first three chapters are clear enough. The 'special matrices' contain, besides several variants of the Wiener matrix mentioned above, also other matrices related to topological indices like Szeged, Hosoya (Z-index) etc. Also compositions of matrices, path matrices, random walk, and transfer matrices. The chapter on 'graphical matrices' is somewhat different because it is about matrices whose elements are subgraphs, not numbers. Of course to 'compute' with them, these subgraphs should be ultimately replaced by some invariant that characterizes them. Several possibilities are given.

The book has the advantage that it is very compact, but it needs to be backed up with a well-stocked library to look up all the references that might be needed for further reading. The title correctly reflects the contents, that is: an enumeration of graph-theoretical matrices with a clear focus on (or even a restriction to) the ones important for mathematical chemistry.

**Submitted by Adhemar Bultheel |

**1 / Jun / 2015