This book presents an account of newer topics, including Szemer'edi's Regularity Lemma and its use; Shelah's extension of the Hales-Jewett Theorem; the precise nature of the phase transition in a random graph process; the connection between electrical networks and random walks on graphs; and the Tutte polynomial and its cousins in knot theory.Expand

The method determines the asymptotic distribution of the number of short cycles in graphs with a given degree sequence, and gives analogous formulae for hypergraphs.Expand

The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the… Expand

Here the authors obtain P(d) asymptotically for all d≤n1/15, where n is the number of vertices, proving as a consequence that γ=3.9±0.1 is obtained.Expand

We consider a random graph process in which vertices are
added to the graph one at a time and joined to a fixed number
m of earlier vertices, where
each earlier vertex is chosen with probability… Expand

We introduce a very general model of an inhomogenous random graph with independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling… Expand