Asymptotic behavior of the number of Eulerian orientations of graphs
  We consider the class of simple graphs with large algebraic connectivity (the second-smallest eigenvalue of the Laplacian matrix). For this class of graphs we determine the asymptotic behavior of the number of Eulerian orientations. In addition, we establish some new properties of the Laplacian matrix, as well as an estimate of a conditionality of matrices with the asymptotic diagonal predominance 
Mikhail Isaev (CMAP)
math.CO