Unique description of chemical structures on hierarchically ordered extended connectivities (HOC procedures). II. Mathematical proofs for the HOC algorithm

Abstract

The basic notions and definitions, necessary for the better understanding of Part I of this series, are presented. The mathematical proof is given for sufficiency of the various HOC procedures for vertex canonical numbering and graph orbit finding.

Determination of Topo-Geometrical Equivalence Classes of Atoms

“…Such a result cannot be guaranteed if one uses only local invariants however discriminant they may be, and although this was pointed out by Trucco and Carhart, some incomplete methods have still been proposed recently. , Both methods fail in some cases of highly symmetric peri-condensed cycles such as the one presented in Figure . However, there exist some algorithms which avoid the problems mentioned by Carhart, either by making a topological classification of the atoms followed by an exhaustive search of the equivalence relationships, like the HOC (hierarchically ordered extended connectivities) procedures, − or by making use of highly elaborated atomic descriptors. From the second group we mention here some algorithms which, after extensive tests, gave correct results: the Gasteiger algorithm, , the partitioning of atoms on the basis of their topological state, the computation of atomic weights by using the layer matrix, or the characterization of the atomic local environment by means of a subspanning tree …”

Spectral Theory of Graphs in Chemistry. 1. Projection Operators and Canonical Numeration of Graph Vertices

“…The literature contains various descriptions of canonical numerations of graph vertices as well as methods of ordering of atoms or functional groups in chemical compounds. [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] Most of these methods deal with the partitioning of the vertices belonging to the molecular graph (MG) or a graph of a more common type into equivalence classes. For this purpose usually vertex invariants of MG are used, which are calculated in terms of various structural characteristics of MG, such as vertex degrees, numbers of the neighbors of the first, second, or higher orders, components of the main eigenvector of the adjacency matrix, and several others (see, for example, refs 9, 10, and [15][16][17][18].…”

“…[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] Most of these methods deal with the partitioning of the vertices belonging to the molecular graph (MG) or a graph of a more common type into equivalence classes. For this purpose usually vertex invariants of MG are used, which are calculated in terms of various structural characteristics of MG, such as vertex degrees, numbers of the neighbors of the first, second, or higher orders, components of the main eigenvector of the adjacency matrix, and several others (see, for example, refs 9, 10, and [15][16][17][18]. A number of canonization algorithms of the adjacency matrix of a simple graph are based on the vertex ordering procedure, which leads to the adjacency matrix corresponding to the minimum or maximum binary code.12-13•20…”