An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Random trees between two walls: Exact partition function

We derive the exact partition function for a discrete model of random trees embedded in a one-dimensional space. These trees have vertices labeled by integers representing their position in the target space, with the SOS constraint that adjacent vertices have labels differing by +1 or -1. A non-trivial partition function is obtained whenever the target space is bounded by walls. We concentrate on the two cases where the target space is (i) the half-line bounded by a wall at the origin or (ii) a segment bounded by two walls at a finite distance. The general solution has a soliton-like structure involving elliptic functions. We derive the corresponding continuum scaling limit which takes the remarkable form of the Weierstrass p-function with constrained periods. These results are used to analyze the probability for an evolving population spreading in one dimension to attain the boundary of a given domain with the geometry of the target (i) or (ii). They also translate, via suitable bijections, into generating functions for bounded planar graphs.Comment: 25 pages, 7 figures, tex, harvmac, epsf; accepted version; main modifications in Sect. 5-6 and conclusio

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

The mixing time of the switch Markov chains: a unified approach

Since 1997 a considerable effort has been spent to study the mixing time of switch Markov chains on the realizations of graphic degree sequences of simple graphs. Several results were proved on rapidly mixing Markov chains on unconstrained, bipartite, and directed sequences, using different mechanisms. The aim of this paper is to unify these approaches. We will illustrate the strength of the unified method by showing that on any $P$-stable family of unconstrained/bipartite/directed degree sequences the switch Markov chain is rapidly mixing. This is a common generalization of every known result that shows the rapid mixing nature of the switch Markov chain on a region of degree sequences. Two applications of this general result will be presented. One is an almost uniform sampler for power-law degree sequences with exponent $\gamma>1+\sqrt{3}$. The other one shows that the switch Markov chain on the degree sequence of an Erd\H{o}s-R\'enyi random graph $G(n,p)$ is asymptotically almost surely rapidly mixing if $p$ is bounded away from 0 and 1 by at least $\frac{5\log n}{n-1}$.Comment: Clarification