1,278 research outputs found
Random graphs from a weighted minor-closed class
There has been much recent interest in random graphs sampled uniformly from
the n-vertex graphs in a suitable minor-closed class, such as the class of all
planar graphs. Here we use combinatorial and probabilistic methods to
investigate a more general model. We consider random graphs from a
`well-behaved' class of graphs: examples of such classes include all
minor-closed classes of graphs with 2-connected excluded minors (such as
forests, series-parallel graphs and planar graphs), the class of graphs
embeddable on any given surface, and the class of graphs with at most k
vertex-disjoint cycles. Also, we give weights to edges and components to
specify probabilities, so that our random graphs correspond to the random
cluster model, appropriately conditioned.
We find that earlier results extend naturally in both directions, to general
well-behaved classes of graphs, and to the weighted framework, for example
results concerning the probability of a random graph being connected; and we
also give results on the 2-core which are new even for the uniform (unweighted)
case.Comment: 46 page
Polygonal Complexes and Graphs for Crystallographic Groups
The paper surveys highlights of the ongoing program to classify discrete
polyhedral structures in Euclidean 3-space by distinguished transitivity
properties of their symmetry groups, focussing in particular on various aspects
of the classification of regular polygonal complexes, chiral polyhedra, and
more generally, two-orbit polyhedra.Comment: 21 pages; In: Symmetry and Rigidity, (eds. R.Connelly, A.Ivic Weiss
and W.Whiteley), Fields Institute Communications, to appea
A Sidetrack-Based Algorithm for Finding the k Shortest Simple Paths in a Directed Graph
We present an algorithm for the k shortest simple path problem on weighted
directed graphs (kSSP) that is based on Eppstein's algorithm for a similar
problem in which paths are allowed to contain cycles. In contrast to most other
algorithms for kSSP, ours is not based on Yen's algorithm and does not solve
replacement path problems. Its worst-case running time is on par with
state-of-the-art algorithms for kSSP. Using our algorithm, one may find O(m)
simple paths with a single shortest path tree computation and O(n + m)
additional time per path in well-behaved cases, where n is the number of nodes
and m is the number of edges. Our computational results show that on random
graphs and large road networks, these well-behaved cases are quite common and
our algorithm is faster than existing algorithms by an order of magnitude.
Further, the running time is far better predictable due to very small
dispersion
Entropy of unimodular Lattice Triangulations
Triangulations are important objects of study in combinatorics, finite
element simulations and quantum gravity, where its entropy is crucial for many
physical properties. Due to their inherent complex topological structure even
the number of possible triangulations is unknown for large systems. We present
a novel algorithm for an approximate enumeration which is based on calculations
of the density of states using the Wang-Landau flat histogram sampling. For
triangulations on two-dimensional integer lattices we achive excellent
agreement with known exact numbers of small triangulations as well as an
improvement of analytical calculated asymptotics. The entropy density is
consistent with rigorous upper and lower bounds. The presented
numerical scheme can easily be applied to other counting and optimization
problems.Comment: 6 pages, 7 figure
Motif counting beyond five nodes
Counting graphlets is a well-studied problem in graph mining and social network analysis. Recently, several papers explored very simple and natural algorithms based on Monte Carlo sampling of Markov Chains (MC), and reported encouraging results. We show, perhaps surprisingly, that such algorithms are outperformed by color coding (CC) [2], a sophisticated algorithmic technique that we extend to the case of graphlet sampling and for which we prove strong statistical guarantees. Our computational experiments on graphs with millions of nodes show CC to be more accurate than MC; furthermore, we formally show that the mixing time of the MC approach is too high in general, even when the input graph has high conductance. All this comes at a price however. While MC is very efficient in terms of space, CC’s memory requirements become demanding when the size of the input graph and that of the graphlets grow. And yet, our experiments show that CC can push the limits of the state-of-the-art, both in terms of the size of the input graph and of that of the graphlets
Effective Scalar Products for D-finite Symmetric Functions
Many combinatorial generating functions can be expressed as combinations of
symmetric functions, or extracted as sub-series and specializations from such
combinations. Gessel has outlined a large class of symmetric functions for
which the resulting generating functions are D-finite. We extend Gessel's work
by providing algorithms that compute differential equations these generating
functions satisfy in the case they are given as a scalar product of symmetric
functions in Gessel's class. Examples of applications to k-regular graphs and
Young tableaux with repeated entries are given. Asymptotic estimates are a
natural application of our method, which we illustrate on the same model of
Young tableaux. We also derive a seemingly new formula for the Kronecker
product of the sum of Schur functions with itself.Comment: 51 pages, full paper version of FPSAC 02 extended abstract; v2:
corrections from original submission, improved clarity; now formatted for
journal + bibliograph
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