Abstract

Much recent attention has focused on the study of models of massive real-world networks like the web graph. The in nite limit graphs of models of these networks satisfy adjacency properties similar to but distinct from the infinite random graph. Each finite graph H generates a corresponding infinite limit graph RH: The isomorphism types of the graphs RH are classified using an ordering of finite graphs named the folding order. We study the endomorphisms and automorphisms of the graphs RH. Using the folding order, we prove that the natural order on the retracts of RH embed all countable orders, and the monoid of all endomorphisms of RH embeds all countable semigroups. We study which partial isomorphisms of RH extend to automorphisms, and prove that all countable groups embed in the automorphism group of RH: We prove that the folding order embeds all finite orders.