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Random graphons and a weak Positivstellensatz for graphs
Author(s) -
Lovász László,
Szegedy Balázs
Publication year - 2012
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.20611
Subject(s) - mathematics , discrete mathematics , combinatorics , multiplicative function , random graph , countable set , graph , ergodic theory , pure mathematics , mathematical analysis
Abstract In an earlier article, the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2‐variable real functions called graphons, random graph models satisfying certain consistency conditions, and normalized, multiplicative and reflection positive graph parameters. In this article we show that each of these structures has a related, relaxed version, which are also equivalent. Using this, we describe a further structure equivalent to graph limits, namely probability measures on countable graphs that are ergodic with respect to the group of permutations of the nodes. As an application, we prove an analogue of the Positivstellensatz for graphs: we show that every linear inequality between subgraph densities that holds asymptotically for all graphs has a formal proof in the following sense: it can be approximated arbitrarily well by another valid inequality that is a “sum of squares” in the algebra of partially labeled graphs. © 2011 Wiley Periodicals, Inc. J Graph Theory