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Enumeration problems on lattices

Ocansey, Evans Doe (2013-03)

Thesis (MSc)--Stellenbosch University, 2013.


ENGLISH ABSTRACT: The main objective of our study is enumerating spanning trees (G) and perfect matchings PM(G) on graphs G and lattices L. We demonstrate two methods of enumerating spanning trees of any connected graph, namely the matrix-tree theorem and as a special value of the Tutte polynomial T(G; x; y). We present a general method for counting spanning trees on lattices in d 2 dimensions. In particular we apply this method on the following regular lattices with d = 2: rectangular, triangular, honeycomb, kagomé, diced, 9 3 lattice and its dual lattice to derive a explicit formulas for the number of spanning trees of these lattices of finite sizes. Regarding the problem of enumerating of perfect matchings, we prove Cayley’s theorem which relates the Pfaffian of a skew symmetric matrix to its determinant. Using this and defining the Pfaffian orientation on a planar graph, we derive explicit formula for the number of perfect matchings on the following planar lattices; rectangular, honeycomb and triangular. For each of these lattices, we also determine the bulk limit or thermodynamic limit, which is a natural measure of the rate of growth of the number of spanning trees (L) and the number of perfect matchings PM(L). An algorithm is implemented in the computer algebra system SAGE to count the number of spanning trees as well as the number of perfect matchings of the lattices studied.

AFRIKAANSE OPSOMMING: Die hoofdoel van ons studie is die aftelling van spanbome (G) en volkome afparings PM(G) in grafieke G en roosters L. Ons beskou twee metodes om spanbome in ’n samehangende grafiek af te tel, naamlik deur middel van die matriks-boom-stelling, en as ’n spesiale waarde van die Tutte polinoom T(G; x; y). Ons behandel ’n algemene metode om spanbome in roosters in d 2 dimensies af te tel. In die besonder pas ons hierdie metode toe op die volgende reguliere roosters met d = 2: reghoekig, driehoekig, heuningkoek, kagomé, blokkies, 9 3 rooster en sy duale rooster. Ons bepaal eksplisiete formules vir die aantal spanbome in hierdie roosters van eindige grootte. Wat die aftelling van volkome afparings aanbetref, gee ons ’n bewys van Cayley se stelling wat die Pfaffiaan van ’n skeefsimmetriese matriks met sy determinant verbind. Met behulp van hierdie stelling en Pfaffiaanse oriënterings van planare grafieke bepaal ons eksplisiete formules vir die aantal volkome afparings in die volgende planare roosters: reghoekig, driehoekig, heuningkoek. Vir elk van hierdie roosters word ook die “grootmaat limiet” (of termodinamiese limiet) bepaal, wat ’n natuurlike maat vir die groeitempo van die aantaal spanbome (L) en die aantal volkome afparings PM(L) voorstel. ’n Algoritme is in die rekenaaralgebra-stelsel SAGE geimplementeer om die aantal spanboome asook die aantal volkome afparings in die toepaslike roosters af te tel.

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