Random Graph Isomorphism

TL;DR: In this paper, the authors propose a framework for learning convolutional neural networks for arbitrary graphs, and demonstrate that the learned feature representations are competitive with state of the art graph kernels and that their computation is highly efficient.

TL;DR: All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.

TL;DR: It is shown that Ω(n) variables are needed for first-order logic with counting to identify graphs onn vertices, equivalent to the (k−1)-dimensional Weisfeiler-Lehman method, and the lower bound is optimal up to multiplication by a constant.

TL;DR: This chapter provides an overview of the common paradigms and results in graph algorithms with an emphasis on the more recent developments in the field, as well as some common optimalization problems on graphs.

Abstract: The classic text for understanding complex statistical probability An Introduction to Probability Theory and Its Applications offers comprehensive explanations to complex statistical problems. Delving deep into densities and distributions while relating critical formulas, processes and approaches, this rigorous text provides a solid grounding in probability with practice problems throughout. Heavy on application without sacrificing theory, the discussion takes the time to explain difficult topics and how to use them. This new second edition includes new material related to the substitution of probabilistic arguments for combinatorial artifices as well as new sections on branching processes, Markov chains, and the DeMoivreLaplace theorem.

TL;DR: The paper sets out to investigate the degree sequences d"1>=d"2>=...>= d"n of random graphs of order n in which the edges are chosen independently and with the same probability p, 0~ arbitrarily slowly, then there is an interval of length C(n) (n@?log ( n@?m)^1^2 which contains the mth highest degree, d"m, of a.e. graph.

TL;DR: It is shown that if G is a simple graph with maximum vertex-degree ϱ, then the chromatic index of G is either ϱ or ϱ + 1, and it is deduced that almost all graphs have Chromatic index equal to their maximum degree.