On the class of graphs with strong mixing properties

We study three mixing properties of a graph: large algebraic connectivity, large Cheeger constant (isoperimetric number) and large spectral gap from 1 for the second largest eigenvalue of the transition probability matrix of the random walk on the graph. We prove equivalence of this properties (in some sense). We give estimates for the probability for a random graph to satisfy these properties. In addition, we present asymptotic formulas for the numbers of Eulerian orientations and Eulerian circuits in an undirected simple graph

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 predominanceComment: arXiv admin note: text overlap with arXiv:1104.304

Automatic enumeration of regular objects

We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These differential equations are then used to determine the initial counting sequence and for asymptotic analysis. The key tool is the scalar product for symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer Sequence

A Dirac type result on Hamilton cycles in oriented graphs

We show that for each \alpha>0 every sufficiently large oriented graph G with \delta^+(G),\delta^-(G)\ge 3|G|/8+ \alpha |G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen. In fact, we prove the stronger result that G is still Hamiltonian if \delta(G)+\delta^+(G)+\delta^-(G)\geq 3|G|/2 + \alpha |G|. Up to the term \alpha |G| this confirms a conjecture of H\"aggkvist. We also prove an Ore-type theorem for oriented graphs.Comment: Added an Ore-type resul