Archive ouverte UNIGE | last documentshttps://archive-ouverte.unige.ch/Latest objects deposited in the Archive ouverte UNIGEengNot Every Uniform Tree Covers Ramanujan Graphshttps://archive-ouverte.unige.ch/unige:12428https://archive-ouverte.unige.ch/unige:12428The notion of Ramanujan graph has been extended to not necessarily regular graphs by Y. Greenberg. We construct infinite trees with infinitely many finite quotients, none of which is Ramanujan. We give a sufficient condition for a finite graph to be covered by such a tree.Fri, 12 Nov 2010 12:58:12 +0100Graph Invariants Related to Statistical Mechanical Models: Examples and Problemshttps://archive-ouverte.unige.ch/unige:12197https://archive-ouverte.unige.ch/unige:12197Spin models and vertex models on graphs are defined as appropriate generalizations of the Ising-Potts model of statistical mechanics. We review some of these state models and the graph functions defined by them. If a graph X represents a knot or a link L in Image 3, we describe models M for which the value ZMX at X of the graph function defined by M depends only on L and not on X.Thu, 21 Oct 2010 14:09:04 +0200Universality and asymptotics of graph counting problems in non-orientable surfaces☆https://archive-ouverte.unige.ch/unige:11879https://archive-ouverte.unige.ch/unige:11879Bender–Canfield showed that a plethora of graph counting problems in orientable/non-orientable surfaces involve two constants tg and pg for the orientable and the non-orientable case, respectively. T.T.Q. Le and the authors recently discovered a hidden relation between the sequence tg and a formal power series solution u(z) of the Painlevé I equation which, among other things, allows to give exact asymptotic expansion of tg to all orders in 1/g for large g. The paper introduces a formal power series solution v(z) of a Riccati equation, gives a non-linear recursion for its coefficients and an exact asymptotic expansion to all orders in g for large g, using the theory of Borel transforms. In addition, we conjecture a precise relation between the sequence pg and v(z). Our conjecture is motivated by the enumerative aspects of a quartic matrix model for real symmetric matrices, and the analytic properties of its double scaling limit. In particular, the matrix model provides a computation of the number of rooted quadrangulations in the 2- dimensional projective plane. Our conjecture implies analyticity of the O(N)- and Sp(N)-types of free energy of an arbitrary closed 3-manifold in a neighborhood of zero. Finally, we give a matrix model calculation of the Stokes constants, pose several problems that can be answered by the Riemann–Hilbert approach, and provide ample numerical evidence for our results.Thu, 23 Sep 2010 14:02:27 +0200