Study on dimer models containing impurities

平面二部图上的完美匹配作为统计力学中的二聚体模型得到了广泛的研究。对于基础图是非二部图的情况，几乎没有已知的结果。在这项研究中，我们研究了通过在平面二部图上增加一些额外的边而得到的图上的完美匹配，如正方形图和六角形图。连接最近的黑色(或白色)顶点的额外边。因此，这类图上的完美匹配可以被认为是某种意义上的含有杂质的完美匹配。我们推广了我们自己的只有一个描述模型的结果，得到了多个杂质存在的情况下的一些结果。但与单一杂质情况相比，这些结果是有限的。

Perfect matchings on planar bipartite graphs have been extensively studied as the dimer models in statistical mechanics. Almost no results are known for the case in which the underlying graphs are non-bipartite. In this study, we investigate the perfect matchings on the graphs which can be obtained by adding some extra edges tothe planar bipartite graphs, such as square graphs and hexagonal graphs. Extra edges connecting nearest black(or white) vertices. Therefore the perfect matchings on such graphs can be considered as perfect matchings containing impurities in some sence.Generalizing our own results which describes the models with only one, we obtained some results on the cases in which multiple impurities exist. But these results are limited when compared to the single impurity case.