在研究矩形的最少个数正方形镶嵌这一组合问题时，Kenyon提出了一个关于平面图的边-生成树密度上界的猜想。该猜想现在被称为Kenyon猜想。通过纽结投影图与符号平图的对应关系，Kenyon猜想可被翻译成纽结理论的语言，即纽结的行列式密度猜想。同时，纽结的行列式密度猜想又与双曲纽结的双曲体积这一低维拓扑中的核心领域密切相关。2015年起，Kenyon猜想逐渐成为研究热点，但离彻底解决还有相当距离。本项目将以Kenyon猜想以及一些相关问题为核心，综合运用图论和低维拓扑中的方法与技巧，研究图（特别是正则图）的生成树计数、纽结的行列式密度、图的边-生成树密度以及图的生成树熵，并拟将Kenyon猜想推广至虚拟纽结上。

Kenyon raised a conjecture on the upper bound of the edge-spanning tree density of a planar graph when studying tiling a rectangle with fewest squares. This conjecture is now named after Kenyon. Through the correspondence between knot diagrams and signed plane graphs, it can be translated into the language of knot theory, which is called the determinant density conjecture of knots. Meanwhile, the determinant density conjecture of knots are closely related to the hyperbolic volume of hyperbolic knots, a core field in low-dimensional topology. Kenyon's conjecture has become a hot topic since 2015, but far from being solved. In this project, we will focus on the Kenyon's conjecture and some related problems. Combining methods and techniques in graph theory and knot theory, we will study the enumeration of spanning trees of graphs (especially regular graphs), the determinant density of knots, the edge-spanning tree density of graphs, and the spanning tree entropy of graphs. Moreover, we expect to generalize the Kenyon's conjecture to virtual knots.