The Atom-Connectivity Matrix

The atom-connectivity matrix, denoted by ACM, has been proposed by Spialter (1963, 1964a, 1964b) for use in computer-oriented chemical nomenclature. This matrix represents the structural formula of a molecule and is given by 

It should also be pointed out that the roots of the atom-connectivity matrix concept go back at least to Balandin (1940), though there are indications that this concept is even older (Randic and Trinajstid, 1994). Balandin constructed property matrices by using the symbols of atoms making up the molecule as diagonal elements and molecular properties such as the interatomic distances (in A), bond dissociation energies (in kcal), and vibrational force constants (in cm x 10 ) as off-diagonal elements. 

The important step in the development of the concept was made by Wheland. In 1946, Wheland (1946) represented molecules as tableaux (connection tables). This Wheland contribution may be regarded as a starting point of the computer-oriented chemical documentation (Trinajstic and Toth, 1986). The Wheland tableau for l-bromo-2-chlorocycloprop-2-ene is given below and is practically identical to the ACM matrix of Spialter. 

Since Wheland did not pursue his idea, this result of his remained practically unknown. Thus, for example, Spialter did not mention the Wheland tableaux in his papers. After Wheland, several authors, located either in industry or at the former National Bureau of Standards (now the National Institute for Standards and Technology) in Washington, D.C., developed methods for computer storage and retrieval of chemical structures (Ray and Kirsch, 1957 Gluck, 1965 Meyer, 1969). Spialter himself was at the Wright-Patterson Air Force Base in Fairborn near Dayton, Ohio. Apparently the need for computer-based systans for chemical documentation was perceived much earlier in applied research than in academic research. 

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Both the adjacency and distance matrices provide information about the connections in the molceular structure, but no additional information such as atom type or bond order. One type of matrix which includes more information, the Atom Connectivity Matrix (ACM), was introduced by Spialtcr and is discussed in Ref, [38]. This approach was eventually abandoned but is listed here because it was quite a unique approach. 

Spialter L (1964) The atom connectivity matrix (ACM) and its characteristic polynomial (ACMCP), J Chem Doc 4 261... 

An analogous quantity, called the bond vertex degree 6f, is calculated from the - atom connectivity matrix C as ... 

The most well-known is the atom connectivity matrix C obtained from multigraphs using the -> conventional bond order jt to represent the adjacency between a pair of vertices ... 

Spialter, L. (1963). The Atom Connectivity Matrix (ACM) and its Characteristic Polynomial (ACMCP) A New Computer-Oriented Chemical Nomenclature. J.Am.Chem.Soc., 85, 2012-2013. 

The bond vertex degree 5 is another local invariant that accounts for atom connectedness and also for bond multiplicity. It is calculated from the atom connectivity matrix C as the sum of row entries [Kier and Hall, 1986] ... 

The adjacency matrix of a multigraph is a variant of the atom connectivity matrix defined as [Janezic, Milicevic et al., 2007]... 

Note that for this molecule the adjacency matrix of the multigraph " A coincides with the atom connectivity matrix C since no aromatic bonds are present in 5-methyl-l,3,4-oxathiazol-2-one. 

Adjacency matrix ofthe multigraph "A for two different canonical structures of styrene and the corresponding adjacency matrix of the general graph A and the atom connectivity matrix C VS, indicates the matrix row sums. 

Spialter, L. (1964b) The atom connectivity matrix characteristic polynomial (ACMCP) and its physico-geometric (topological) significance. 

In 1964, Spialter introduced the atom connectivity matrix (ACM) in an attempt to define algebraically a characteristic polynomial associated... 

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