A technique for determining the symmetry properties of molecular graphs

Abstract

A procedure has been devised to determine the automorphism partition and the automorphism group for nondirected graphs. The general approach is to reveal the permutational symmetry by means of its systematic destruction. Symmetry decomposition pathways are created by numerically weighting specific series of vertices. Each pathway leads to a discrete ordered vector. Each pair of pathways that match reveals an element of the automorphism group. Results are presented for two molecular graphs. Further potential applications of this approach to problems of isomorphism, canonical labeling, and structural indexing are discussed.

Computational Techniques for the Automorphism Groups of Graphs

“…In symbols, J-'cs", x s"2 x ... x s"m (5) The number of times eq 3 needs to be checked reduces significantly as there are more equivalence classes and less number of vertices in each class. It can be seen that the number of multiplication operations for the exhaustive search of automophisms in this case is T(nx,n2,...,nm) = 2n3 nx\n¿n3L.nJ (6) To illustrate consider a graph which contains 16 vertices and suppose that the vertices are partitioned into 3 equivalence classes, the first class containing 8 vertices, the second and third containing 4 vertices each. The total number of multiplication operations reduce from 1.714 X 1017 to 1.902 X 10" as a result of vertex partitioning.…”

“…This states that if g¡ ^e £ G, then g,2, g,•3,..., gf ^e £ G. For all i = 1,2, ..., m, the powers of g¡ are constructed until the 2 (Figure 1) <0.03 s 4.52 s 6 3 (Figure 1, Peterson) 2 min, 10 s 3 h, 34 min, 56 s 120 4 (Figure 1, cube) 13s 19 min, 49 s 48 5 (Figure 1, pentagon in 16.6 s 1 h, 1 min, 43 s 20 pentagon) 6 (Figure 1, hexagon in 39 min, 29 s >137 h 24 hexagon) anthracene 0.13 s 3 min, 28.7 s 4 " All CPU times were measured on an IBM RS/6000 580 workstation. 6 The algorithm from Razinger et al1 identity is reached and the set is extended to encompass these new elements. Now the program can be restarted with the current permutation as the last generated permutation.…”

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

“…Definition 7: Two cycles, C,, and C,,,,,, in a graph are said to be vertex equivalent when: card C,, (u) = card Ca,s,a,( V ) (9) u, E C,,(V) = ui, E C,.,.,. (V) (10) hold for each…”

Symmetry properties of chemical graphs. VI. Isomerizations of octahedral complexes

“…It exploits the utility of the adjacency matrix of a graph in the role of a projection operator. A detailed description of this approach has recently been published [20] so we shall limit ourselves here to a brief account of those features which have special reference to the analysis of our 15-vertex graph.…”