Vine copulas can efficiently model multivariate probability distributions. This paper focuses on a more thorough understanding of their structures, since in the literature, vine copula representations are often ambiguous. The graph representations include the original, cherry and chordal graph sequence structures, which we show equivalence between. Importantly we also show a new result, namely that when a perfect elimination ordering of a vine structure is given, then it can always be uniquely represented with a matrix. O. M. N\'apoles has shown a way to represent vines in a matrix, and we algorithmify this previous approach, while also showing a new method for constructing such a matrix, through cherry tree sequences. We also calculate the runtime of these algorithms. Lastly, we prove that these two matrix-building algorithms are equivalent if the same perfect elimination ordering is being used.

翻译：藤蔓连接函数能有效建模多元概率分布。本文旨在更深入地理解其结构，因为现有文献中对藤蔓连接结构的表示常存在歧义。图表示包括原始结构、樱桃结构及弦图序列结构，我们证明这些结构是等价的。重要地，我们还证明了新结论：当给定藤蔓结构的完美消除排序时，该结构总能用矩阵唯一表示。O. M. Nápoles曾提出用矩阵表示藤蔓的方法，我们对该方法进行了算法化改进，同时通过樱桃树序列提出了构建此类矩阵的新方法。此外，我们计算了这些算法的运行时间。最后，我们证明：若使用相同的完美消除排序，则这两种矩阵构建算法是等价的。