Asymptotic behavior of the number of Eulerian orientations of graphs
Mikhail Isaev (CMAP)
TL;DR
This paper investigates the asymptotic number of Eulerian orientations in simple graphs with high algebraic connectivity, revealing new properties of the Laplacian matrix and providing estimates related to matrix conditionality.
Contribution
It determines the asymptotic behavior of Eulerian orientations for graphs with large algebraic connectivity and introduces new properties and estimates of Laplacian matrices.
Findings
Asymptotic formula for Eulerian orientations in high-connectivity graphs
New properties of Laplacian matrices related to graph orientations
Estimates on the conditionality of matrices with asymptotic diagonal dominance
Abstract
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
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