On "the matching polynomial of a polygraph"

In this note we give an explanation for two phenomena mentioned in the concluding remarks of “The matching polynomial of a polygraph” by Babić et al. The following results are obtained: \ud 1.\ud Although three matrices for given polygraphs defined in the above article in general have different orders, they determine the same recurrence relations for the matching polynomial of these polygraghs.\ud 2.\ud Under certain symmetry conditions, the order of the recurrence relations can be reduced by almost a half

Pojednostavljeni račun sparivanja u poligrafovima

Matching polynomial and perfect matchings for fasciagraphs, rotagraphs and twisted rotagraphs are treated in the paper. Classical transfer matrix approach makes it possible to get recursions for matching polynomial and perfect matchings, but the order of the matrix grows exponentially in the number of the linking edges between monographs. Novel transfer matrices are introduced whose order is much lower than that in classical transfer matrices. The virtue of the method introduced is especially pronounced when two or more linking edges end in the same terminal vertex of a monograph. An example of a polyacene polygraph with extended pairings is given where a novel matrix has only 16 entries as compared to 65536 entries in the classical transfer matrix. However, all pairings are treated here on equal footing, but the method introduced can be applied to selected types of pairings of interest in chemistry.U radu se razmatraju polinomi sparivanja i savršena sparivanja u fascia- i rotagrafovima te izvijenim rotagrafovima. Iako klasični postupak transfer matrice omogućava izvođenje rekurzija za polinom sparivanja i savršena sparivanja, red ove matrice eksponencijalno raste s brojem veza me|u monografovima. Ovdje su uvedene nove transfer matrice čiji je red mnogo ni`i od onoga za klasične transfer matrice, i to posebice kada jedna ili više veza me|u monografovima završava u jednom te istom čvoru. Postupak je ilustriran na primjeru poliacenskih poligrafova gdje ovdje uvedena matrica ima samo 16 elemenata u usporedbi s 65536 elemenata klasične transfer matrice. Iako se ovdje uvedeni postupak primjenjuje istovremeno na sva moguća sparivanja u poligrafovima, on je otvoren za primjenu na odabrana sparivanja od posebnoga kemijskoga interesa