石墨烯是二维结构的单层石墨，它是由碳原子组成的六角蜂巢状二维晶格结构。自从被发现以来，受到了科学家们的高度关注。石墨烯拥有很高的电子迁移率、十分大的比表面积、较小的质量密度、很高的化学惰性及热稳定性等特征。此外，石墨烯还具有良好的磁输运特性，例如弱局域化抗磁性、室温下的量子霍尔效应（quantum Hall effect）和量子电气力学现象等。也因为其特殊的物理、化学及机械性能，使得石墨烯相关材料有希望广泛应用于自旋电子学、纳米器件、信息及能源领域。本文主要内容如下：论文第一章，简单介绍了石墨烯的物理性质、制备方法和广泛应用。第二章介绍了一种非微扰的数值技术：行列式量子蒙特卡罗方法（determinant quantum Monte Carlo method ：DQMC）及其在数值计算上的优势。论文第三章，利用行列式量子蒙特卡罗方法研究了施加应力之后具有锯齿型边缘的石墨烯量子点的边缘磁性，这是本文的重点。文中我们给出了具有锯齿型和扶手椅型边界的石墨烯量子点晶格结构和模型，计算表明锯齿型边缘磁化率随着温度的降低而增加，呈现一种铁磁性行为，而扶手椅型边缘磁化率随着温度的降低而降低；应变和哈伯德相互作用对锯齿型边缘磁化率具有明显的加强作用；当电子填充远离半满时低温区的锯齿型磁化率略有减小；经过大量数值模拟计算，给出了一组实验上探测石墨烯纳米点边缘磁性的最佳参数值：U=2.3|t|，\Delta t=0.20|t|。论文第四章，利用约束路径蒙特卡罗方法研究了定义在三角晶格上的t-U-V哈伯德模型的基态配对关联性质。研究发现不同的电子配对对称性，在不同的电子填充区占据主导地位。fn-波对称关联在电子半满附近占优，当电子填充远离半满时d+id波配对关联开始占据主导，而当电子浓度继续降低到范霍夫奇点以下时，f波配对关联开始变得更强。这些不同的现象是由于电子关联和几何阻挫的相互作用，以及反铁磁关联和铁磁涨落之间的竞争导致的。我们的发现说明了超导和磁性具有重要的渊源，对于理解 Na_{x}CoO_{2}.H_{2}O和有机化合物超导体等的性质提供了重要的信息。

Graphene is a two-dimensional structure of monolayer graphite, which is made of carbon atoms on a hexagonal honeycomb lattice structure. Graphene has been highly concerned by the whole world since it was discovered. Graphene has high electron mobility, large specific surface area, smaller mass density, chemical inertness and high thermal stability, etc. In addition, graphene also has good magnetic transport properties, such as weak localization diamagnetism, the quantum hall effect at room temperature and quantum electrodynamics, etc. Because of its special properties in physical, chemical and mechanical, graphene has been widely used in spintronics, nano-devices, information and energy fields. This article mainly writing are arranged as following:Firstly, the physical properties of graphene and its possible widely application are introduced.Secondly, a brief introduction on an unbiased numerical calculation method named determinant quantum Monte Carlo method (DQMC) is introduced.In the third Charpter, we study the temperature dependent magnetic susceptibility of a strained graphene quantum dot by using the determinant quantum Monte Carlo method. Within the Hubbard model on a honeycomb lattice, our unbiased numerical results show that a relative small interaction $U$ may lead to a edge ferromagnetic like behavior in the strained graphene quantum dot, and a possible room temperature transition is suggested. Around half filling, the ferromagnetic fluctuations at the zigzag edge is strengthened both markedly by the on-site Coulomb interaction and the strain, especially in low temperature region. The resultant strongly enhanced ferromagnetic like behavior may be important for the development of many applications.In the forth Charpter, we study the ground state paring correlations in the t-U-V Hubbard model on the triangular lattice, by using the constrained path quantum Monte carlo method. It is shown that pairings with various symmetries dominate in different electron filling regions. The pairing correlation with fn-wave symmetry dominates over other pairings around half fillings, and as the electron filling decreases away from the half filling, the d+id-wave pairing correlation tends to dominate. As the electron filling is bellow the Van Hove singularity, the f-wave pairing dominates. These crossovers are due to the interplay of electronic correlation and geometric frustration, associating with the competition between the antiferromagnetic correlations and ferromagnetic fluctuations. Our findings reveal the possible magnetic origin of superconductivity, and also provide useful information for the understanding of superconductivity in Na_{x}CoO_{2}.H_{2}O and the organic compounds.