In the present book we have discussed five classes of graph-theoretical matrices: adjacency matrices, incidence matrices, distance matrices, special matrices and graphical matrices. A total of 130 graph-theoretical matrices, which we regard as important in contemporary chemical graph theory, have been considered. They have found a wide range of applications - they are used to generate many kinds of molecular descriptors increasingly employed in molecular modeling, to generate walks and random walks, which have so far found much use outside chemistry and are used modestly in chemistry to study the complexity of molecules and chemical reactions, to generate and enumerate isomers and valence-bond structures, to start building and to filter virtual combinatorial libraries that are so important for rational preparation of practically any desired compound. We hope that this exposition may stimulate some readers to study these matrices in more detail, since the polynomials, spectra and properties of many of them are still poorly defined. Besides these properties, one needs to know their computational and combinatorial properties in order to establish the range of their applicability in chemistry. It is also of interest to find which if any of these matrices besides the Hückel matrix [2,145-148] and the transfer matrix via the conjugated-circuits model [363], may be applicable in quantum chemistry [ e.g ., 364 ] . A very stimulating article by Klein [365] on graph-theoretically formulated electronic-structure theory explores these ideas. Still other graph-theoretical matrices e.g., for the Pauling-Wheland VB (or Heisenberg) model or for the Hubard model arise implicitly and the wonderful world of graph-theoretical matrices is open to further, hopefully fruitful, exploration.