Symmetries of graphs intervene in diverse applications, ranging from enumeration to graph structure compression, to the discovery of graph dynamics (e.g., inference of the arrival order of nodes in a growing network). It has been known for some time that ErdősRényi graphs are asymmetric with high probability, but it is also known that real world graphs (web, biological networks) have a signific...

We give asymptotic upper and lower bounds for the diameter of almost every r-regular graph on n vertices (n~). Though random graphs of various types have been investigated extensively over the last twenty years, random regular graphs have hardly been studied. The reason for this is that until recently there was no formula for the asymptotic number of labelled r-regular graphs of order n. Such a...

We consider the problem of finding optimal description for general unlabeled graphs. Given a probability distribution on labeled graphs, we introduced in [4] a structural entropy as a lower bound for the lossless compression of such graphs. Specifically, we proved that the structural entropy for the Erdős–Rényi random graph, in which edges are added with probability p, is ` n 2 ́ h(p)−n log n+O(...

We consider several classes of simple graphs as potential models for information diffusion in a structured population. These include biases cycles, dual circular flows, partial bipartite graphs and what we call 'single-link' graphs. In addition to fixation probabilities, we study structure parameters for these graphs, including eigenvalues of the Laplacian, conductances, communicability and exp...

ABSTRACT. A number of recent works have used a variety of combinatorial constructions to derive Tanner graphs for LDPC codes and some of these have been shown to perform well in terms of their probability of error curves and error floors. Such graphs are bipartite and many of these constructions yield biregular graphs where the degree of left vertices is a constant c+1 and that of the right ver...

Plan graphs are commonly used in planning to help compute heuristic “distance” estimates between states and goals. A few authors have also attempted to use plan graphs in probabilistic planning to compute estimates of the probability that propositions can be achieved and actions can be performed. This is done by propagating probability information forward through the plan graph from the initial...

We determine the probability tresholds for the existence of infinite paths in random shift graphs.

We prove that with high probability maximum sizes of induced forests in dense binomial random graphs are concentrated two consecutive values.

Let G denote a graph and let K be a subset of vertices that are a set of target vertices of G. The Kterminal reliability of G is defined as the probability that all target vertices in K are connected, considering the possible failures of non-target vertices of G. The problem of computing K-terminal reliability is known to be #P-complete for polygon-circle graphs, and can be solved in polynomial...

This paper describes the study of a special class of 4-regular plane graphs which are Hamiltonian. These graphs are of special interest in knot theory. An algorithm is presented that randomly generates such graphs with n vertices with a fixed (and oriented) Hamiltonian cycle in O(n) time. An exact count of the number of such graphs with n vertices is obtained and the asymptotic growth rate of t...