Matrix and graph representations of vine copula structures

Vine copulas can efficiently model a large portion of probability distributions. This paper focuses on a more thorough understanding of their structures. We are building on well-known existing constructions to represent vine copulas with graphs as well as matrices. The graph representations include the regular, cherry and chordal graph sequence structures, which we show equivalence between. Importantly we also show 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 them in a matrix, and we algorithmify this previous approach, while also showing a new method for constructing such a matrix, through cherry tree sequences. Lastly, we prove that these two matrix-building algorithms are equivalent if the same perfect elimination ordering is being used.