On the number of Euler trails in directed graphs - Jonsson, Jakob (MATHEMATICA SCANDINAVICA, 2002)
Let $G$ be an Eulerian digraph with all in- and out-degrees equal to 2, and let $π$ be an Euler trail in $G$. We consider an intersection matrix $\boldsymbol {L}(π)$ with the property that the determinant of $\boldsymbol{L}(π) + \boldsymbol{I}$ is equal to the number of Euler trails in $G$; $\boldsymbol{I}$ denotes the identity matrix. We show that if the inverse of $\boldsymbol{L}(π)$ exists, then $\boldsymbol{L}^{-1}(π) = \boldsymbol{L}(σ)$ for a certain Euler trail $σ$ in $G$. Furthermore, we use properties of the intersection matrix to prove some results about how to divide the set of Euler trails in a digraph into smaller sets of the same size.