No abstract

In a random linear graph, vertices are points on a line, and pairs of vertices are connected, independently, with a link probability that decreases with distance. We study the problem of reconstructing the linear embedding from the graph, by recovering the natural order in which the vertices are placed. We propose an approach based on the spectrum of the graph, using recent results on random matrices. We demonstrate our method on a particular type of random linear graph. We recover the order and give tight bounds on the number of misplaced vertices, and on the amount of drift from their natural positions.

In this paper we work out the theory of fractional isomorphism of graphons as a generalization to the classical theory of fractional isomorphism of finite graphs. The generalization is given in terms of Markov operators on a Hilbert space and it is characterized in terms of iterated degree distributions, homomorphism density of trees, weak isomorphism of a conditional expectation with respect to invariant sub-σ-algebras and isomorphism of certain quotients of given graphons. Our proofs use a weak version of the mean ergodic theorem, and correspondences between objects such as Markov projections, sub-σ-algebras, conditional expectation, etc. That also provides an alternative proof for the characterizations of fractional isomorphism of graphs without the use of Birkhoff-von Neumann Theorem.

We study graphon counterparts of the chromatic and the clique number, the fractional chromatic number, the b-chromatic number, and the fractional clique number. We establish some basic properties of the independence set polytope in the graphon setting, and duality properties between the fractional chromatic number and the fractional clique number. We present a notion of perfect graphons and characterize them in terms of induced densities of odd cycles and its complements.