J. Chem. In$ Comput. Sci. 1990, 30, 160-163
160
ACKNOWLEDGMENT
Table I. Enumeration of Polvhexes with UD to h = 16
h
planar polyhexes planar with holes ______ CPU time polyhexes single multiple without holes hole holes days h min
1
1
2 3 4
I 3 7 22 81 33 1 1435 6 505 30086 141 229 669 584 3198256 15367577 74207910 359863778
5 6 7 8 9
IO 11
12 13 14 15 16
1
5 43 283 1954 12363 76283 453946 2641506
1 11
149 1618
3 18 91
3 17 14 3 7
1 8 40 24 4 4 13 24
s
0.44 0.44 0.44 0.44 0.60 1.26 4.67 20.87 40.13 12.90 52.17 1.80 32.50 24.76 34.70 33.69
tween the terms polyhex and benzenoid. Our algorithm for enumeration of polyhexes is based on the DAST (the dualist angle-restricted spanning tree) code.I3 This code is founded on the graph-theoretical concept of the weighted spanning tree of d ~ a 1 i s t . lA~ computer program for enumerating polyhex hydrocarbons using our algorithm is detailed elsewhere.g Counts of planar polyhex hydrocarbons with and without holes with up to h = 16 are given in Table I. In the table we also give CPU time needed to complete computation for each h. For example, the required CPU time for the enumeration of polyhexes with 16 hexagons was 91 days, 7 h, 24 min, and 33.69 s. Computations have been carried out on the Siemens PCD3D (20 MHz, 386-AT). We deliberately did not use our supercomputer to show that these types of combinatorial computations can be completed on the personal computer, if one is willing to spend some time to design carefully an efficient computational algorithm and if one can spare a PC for continuous computations over a long period of time.
This work was supported in part by the German-Yugoslav Scientific Cooperation Program. Financial support from the Internationales Biiro, KernforschungsanlageJiilich, and from the Croatian Science Fund is gratefully acknowledged. We are thankful to reviewers for their critical but helpful comments. REFERENCES AND NOTES ( I ) ToSif, R.; KovaEeviE, M. Generating and Counting Unbranched Catacondensed Benzenoids. J . Chem. InJ Comput. Sci. 1988, 28, 29-31.
(2) Brunvoll, J.; Cyvin, B. N.; Cyvin, S. Enumeration and Classification of Coronoid Hydrocarbons. J. Chem. InJ Cornput. Sci. 1987, 27, 14-21. (3) Brunvoll, J.; Cyvin, B. N.; Cyvin, S. Enumeration and Classification of Benzenoid Hydrocarbons. 2. Symmetry and Regular Benzenoids. J. Chem. InJ Comput. Sci. 1987, 27, 171-177. (4) Cyvin, S.; Brunvoll, J.; Cyvin, B. N . Distribution of K,the Number of Kekult Structures, in Benzenoid Hydrocarbons: Normal Benzenoids with K to 110. J. Chem. InJ Compuf.Sci. 1989, 29, 74-90. (5) Ciosiowski, J. Computer Enumeration of Polyhexes Using the Compact Naming Approach. J . Compur. Chem. 1987.8, 906-915. ( 6 ) Harary, F.; Palmer, E. M. Graphical Enumeration; Academic: New York, 1973. (7) Balasubramanian, K.; Kauffman, J. J.; Koski, W. S.; Balaban, A. T. Graph Theoretical Characterization and Computer Generation of Certain Carcinogenic Benzenoid Hydrocarbons and Identification of Bay Regions. J . Comput. Chem. 1980, I , 149-157. (8) Muller, W. R.; Szymanski, K.; Knop, J. V.; NikoliE, S.; TrinajstiE, N. On the Enumeration and Generation of Polyhex Hydrocarbons. J. Comput. Chem. (in press). (9) TrinajstiE, N. Chemical Graph Theory; CRC: Boca Raton, FL, 1983; Vol. 1, Chapter 3. ( I O ) TrinajstiE, N. On the Classification of Polyhex Hydrocarbons. J . Math. Chem. (in press). ( I I ) Harary, F. Graph Theory; Addison- Wesley: Reading, MA, 197 1 ; second printing. (12) Knop, J. V.; Muller, W. R.; Szymanski, K.; TrinajstiE, N. On the Enumeration of 2-Factors of Polyhexes. J. Compuf. Chem. 1986, 7, 547-564. ( 1 3) Knop, J. V.; Muller, W. R.; Szymanski, K.; NikoliE, S.; TrinajstiC, N. Computer-oriented Molecular Codes. I n Computafional Chemical Graph Theory; Rouvray, D. H., Ed.; Nova: New York, 1990 pp 9-32. (14) NikoliE, S.; TrinajstiE, N.; Knop, J. V.; Muller, W. R.; Szymanski. K. On the Concept of the Weighted Spanning Tree of Dualist. J. Math. Chem. (in press).
Molecular Topological Index WOLFGANG R. MULLER, KLAUS SZYMANSKI, JAN V . KNOP, and NENAD TRINAJSTIt*J Computer Centre, The Heinrich Heine University, 4000 Diisseldorf, Federal Republic of Germany Received November 13, 1989 T h e molecular topological index ( M T I ) has been systematically tested for counterexamples (two o r m o r e nonisomorphic structures with t h e s a m e MTI number). T h e analysis was carried out for alkane trees with u p t o 16 atoms. T h e search for counterexamples was positive: T h e first pair of alkane trees with identical MTI numbers was found in t h e octane family. T h e m o r e disturbing finding was t h a t two nonisomorphic alkane trees of different sizes m a y also possess the s a m e M T I value. A n attempt to redefine t h e MTI in terms of only the distance matrix and t h e valency matrix was abortive.
Schultz' has recently introduced in this journal a novel topological (or, more correctly, graph-theoretical) descriptor for characterization of alkanes by an integer. This descriptor was named by its originator the molecular topological index (MTI). The MTI appears to be an attractive graph-theoretical descriptor that is easy to compute and has structural signif*Permanent address: The Rugjer BojkoviE Institute. P. 0. Box 1016, 41001 Zagreb, Croatia, Yugoslavia.
icance. The important questions, Are the MTI numbers unique?, and, if the answer is positive, What are the smallest graphs for which MTI is not unique?, however, were not considered by Schultz. He only stated that the MTI is a highly discriminative descriptor. In this paper we will try to answer these questions. The MTI is based on the adjacency ( N X N)matrix A, the distance ( N X N) matrix D, and the valency (1 X N) matrix c of a n alkane. The sum of the elements ei ( i = I , 2, ..., N)
0095-2338/90/ 1630-01 60$02.50/0 0 I990 American Chemical Society
J . Chem. In& Comput. Sci.. Vol. 30, No. 2, 1990 161
MOLECULAR TOPOLOGICAL INDEX
Table I. Duplicate MTI Values for Nonane and Decane Trees"
MTI
i
2L2
MTI
32101000
MTI
i
242
:
\
11 2 0 0 0 0 0
260
MTI
31 1 0 1 0 1 0
i
260
L
L1110000
Figure 1. Two pairs of octane trees with degenerate MTI values.
Beneath each tree its N-tuple code is also given.
N-tuple code (a) Nonanes 4 2 100 1000
MTI
42101 0000
318 318
322 100000 411010100
332 332
412001000 421100000
334 334
321200000 41 1011000
348 348
3 12 101000 321100100
354 354
312110000 321 110000
370 370
(b) Decanes 4200200100 4230000000
402 402
4210020000 4220010000
414 414
3220020000 4210010100
420 420
4110101010 4200120000
434 434
3220011000 4212000000
446 446
Figure 2. Triplet of decane trees with the same MTI value. Beneath each tree its N-tuple code is also given.
3221000100 4122000000
450 450
of the product of matrices u(A + D)gives the molecular topological index:
4121001000 4121010000
456 456
4120011000 4200111000
460 460
3211010100 4113000000
464 464
3211011000 4211100000
472 472
4112001000 4121100000
476 476
3121101000 3211101000
484 484
3211001100 4112100000
488 488
3112101000 3121100100
500 500
3112110000 321111
