By Haruo Hosoya (auth.), Danail Bonchev, Ovanes Mekenyan (eds.)
The development in laptop expertise over the past 10-15 years has enabled the functionality of ever extra designated quantum mechanical calculations concerning constitution and interactions of chemicals. besides the fact that, the qualitative versions concerning digital constitution to molecular geometry haven't advanced on the similar velocity. there's a carrying on with desire in chemistry for easy suggestions and qualitatively transparent photographs which are additionally quantitatively similar to ab initio quantum chemical calculations. Topological tools and, extra in particular, graph concept as a fixed-point topology, supply in precept an opportunity to fill this hole. With its greater than a hundred years of functions to chemistry, graph conception has confirmed to be of important value because the so much normal language of chemistry. The explosive improvement of chemical graph thought over the last two decades has more and more overlapped with quantum chemistry. along with contributing to the answer of assorted difficulties in theoretical chemistry, this improvement exhibits that topology is an underlying precept that explains the good fortune of quantum mechanics and is going past it, hence promising to endure extra fruit within the future.
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Chern. Soc. 102, 1837-1848. , and Sachs, H. (1980) Spectra of Graphs: Theory and Application, Academic Press, Berlin. J. and Gutman, I. (1988) Kekule Structures in Benzenoid Hydrocarbons, Lecture Notes in Chemistry, 46, Springer, Berlin. J. and Gutman, I. ) (1990) Advances in the Theory of Benzenoid Hydrocarbons, Topics in Current Chemistry, 153, Springer, Berlin. D. and Prelog, V. (1968) "Ligand Reorganization in the Trigonal Bipyramid", Angew. Chern. IntI. Ed. Engl. 7, 725-726. Engel, P. (1982) "On the Enumeration of Polyhedra", Discrete Math.
0-1 2x _(x 2 -1) -xO (0-1) x (4-10) =0 and we have 6 3 - (x4 -5x 3 +3)6 2 + (x4-3x2+3)6 - 1 = O. which gives the corresponding recurrence formula for the matching polynomials of the three series of graphs as Fn = (x4-5x 3 +3)Fn_l - (x4 -3x 2+3)F n_2 + F n-3 (4-11) Next consider another type of I-dimensional hexagonal animals. which grow up in such a manner that addition of a hexagonal cell is randomly chosen from linear (I). helical (H). or zigzag (F) mode (Randic et al.. 1989). Linear growth (I) Helical (H) Zigzag (F) The characteristic and matching polynomials of this aperiodic polymer network can be easily obtained by the step-by-step multiplication of the three kinds of transfer matrices just in the same order as the hexagonal cells grow by combining the above three modes.
The characteristic polynomials of the topomers can be expressed in terms of the characteristic polynomials of their subunits following the ideas of Heilbronner. S3 Thus, the characteristic polynomials of topomers Sand T from Figure 7 are given by, 48 N. TRINAJSTIC ET AL. P(S;x) = P(A;x) P(B;x) - P(A-r;x) P(B-u;x) - P(A-s;x) P(B-v;x) + P(A-r-s;x) P(B-u-v;x) - 2[I. P(A-Prs;x)] [I. P(B-puv;x)] (29) P(T;x) = P(A;x) P(B;x) - P(A-r;x) P(B-v;x) - P(A-s;x) P(B-u;x) + P(A-r-s;x) P(B-u-v;x) - 2[I. P(A-Prs;x)] [I.