Showing papers on "Quartic graph published in 2008"

Graph summarization with bounded error

Abstract: We propose a highly compact two-part representation of a given graph G consisting of a graph summary and a set of corrections. The graph summary is an aggregate graph in which each node corresponds to a set of nodes in G, and each edge represents the edges between all pair of nodes in the two sets. On the other hand, the corrections portion specifies the list of edge-corrections that should be applied to the summary to recreate G. Our representations allow for both lossless and lossy graph compression with bounds on the introduced error. Further, in combination with the MDL principle, they yield highly intuitive coarse-level summaries of the input graph G. We develop algorithms to construct highly compressed graph representations with small sizes and guaranteed accuracy, and validate our approach through an extensive set of experiments with multiple real-life graph data sets.To the best of our knowledge, this is the first work to compute graph summaries using the MDL principle, and use the summaries (along with corrections) to compress graphs with bounded error.

On quartic half-arc-transitive metacirculants

Abstract: Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ? and ?, where ? is (m,n)-semiregular for some integers m?1, n?2, and where ? normalizes ?, cyclically permuting the orbits of ? in such a way that ? m has at least one fixed vertex. A half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. In this article quartic half-arc-transitive metacirculants are explored and their connection to the so called tightly attached quartic half-arc-transitive graphs is explored. It is shown that there are three essentially different possibilities for a quartic half-arc-transitive metacirculant which is not tightly attached to exist. These graphs are extensively studied and some infinite families of such graphs are constructed.

An Expansion Tester for Bounded Degree Graphs

Abstract: We consider the problem of testing graph expansion (eithervertex or edge) in the bounded degree model [10]. We give aproperty tester that given a graph with degree bound d, anexpansion bound α, and a parametere> 0, accepts the graph with high probabilityif its expansion is more than α, and rejects it withhigh probability if it is e-far from any graphwith expansion α' with degree bound d,where α' 0.

Determinants and Longest Cycles of Graphs

Abstract: We consider the Hamiltonian cycle problem on a given graph $G$. With such a graph we can associate a family $\mathcal{F}$ of probability transition matrices of Markov chains whose entries represent the probabilities of traversing corresponding arcs of the graph. When the underlying graph is Hamiltonian, we show the transition probability matrix induced by a Hamiltonian cycle maximizes—over $\mathcal{F}$—the determinant of a matrix that is a rank-one correction of the generator matrix of a Markov chain. In the case when the graph does not possess a Hamiltonian cycle, the above maximization yields a transition matrix of a chain with a longest simple cycle (in $G$) comprising that chain's unique ergodic class. These problems also have analogous eigenvalue interpretations.

Computing monomer-dimer systems through matrix permanent.

Abstract: The monomer-dimer model is fundamental in statistical mechanics. However, it is $#P$-complete in computation, even for two-dimensional problems. A formulation for the partition function of the monomer-dimer system is proposed in this paper by transforming the number of all matchings of a bipartite graph into the number of perfect matchings of an extended bipartite graph, which can be given by a matrix permanent. Sequential importance sampling algorithm is applied to compute the permanents. For two-dimensional lattice with periodic condition, the monomer-dimer constant is known as ${h}_{2}=0.662798972834$. We obtain $0.6627\ifmmode\pm\else\textpm\fi{}0.0002$ for our approximation, which shows the robustness and the efficiency of the algorithm. For three-dimensional problem, our numerical result is $0.7847\ifmmode\pm\else\textpm\fi{}0.0014$, which agrees with the best known bounds.

Alternative similarity functions for graph kernels

Abstract: Given a bipartite graph of collaborative ratings, the task of recommendation and rating prediction can be modeled with graph kernels. We interpret these graph kernels as the inverted squared Euclidean distance in a space defined by the underlying graph and show that this inverted squared Euclidean similarity function can be replaced by other similarity functions. We evaluate several such similarity functions in the context of collaborative item recommendation and rating prediction, using the exponential diffusion kernel, the von Neumann kernel, and the random forest kernel as a basis. We find that the performance of graph kernels for these tasks can be increased by using these alternative similarity functions.

Affinity Measures Based on the Graph Laplacian

Abstract: Several language processing tasks can be inherently represented by a weighted graph where the weights are interpreted as a measure of relatedness between two vertices Measuring similarity between arbitary pairs of vertices is essential in solving several language processing problems on these datasets Random walk based measures perform better than other path based measures like shortest-path We evaluate several random walk measures and propose a new measure based on commute time We use the psuedo inverse of the Laplacian to derive estimates for commute times in graphs Further, we show that this pseudo inverse based measure could be improved by discarding the least significant eigenvectors, corresponding to the noise in the graph construction process, using singular value decomposition

Distributed Lyapunov Functions in Analysis of Graph Models of Software

Abstract: In previous works, the authors introduced a framework for software analysis, which is based on optimization of Lyapunov invariants. These invariants prove critical software properties such as absence of overflow and termination in finite time. In this paper, graph models of software are introduced and the software analysis framework is further developed and extended on graph models. A distributed Lyapunov function is assigned to the software by assigning a Lyapunov function to every node on its graph model. The global decremental condition is then enforced by requiring that the Lyapunov functions on each node decrease as transitions take place along the arcs. The concept of graph reduction and optimality of graphs for Lyapunov analysis is briefly discussed.

Degree sum conditions in graph pebbling.

Abstract: Given a graph G on n vertices and a distribution, D, of pebbles on the vertices of G, we define a pebbling move to be the removal of two pebbles from a given vertex and the placement of one on an adjacent vertex. If D has n pebbles and if after a sequence of pebbling moves we can place a pebble on any specified vertex then we call G Class 0. We give a sufficient degree sum condition for G to be Class 0.

A Characterization of the degree sequences of 2-trees

Abstract: A graph G is a 2-tree if G = K3, or G has a vertex v of degree 2, whose neighbors are adjacent, and G- v is a 2- tree. A characterization of the degree sequences of 2-trees is given. This characterization yields a linear-time algorithm for recognizing and realizing degree sequences of 2-trees. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:191-209, 2008 This research was initiated at The 21st Bellairs Winter Workshop on Computational Geometry, January 27–February 3, 2006.

Coarse grained parallel algorithms for graph matching

Abstract: Parallel graph algorithm design is a very well studied topic. Many results have been presented for the PRAM model. However, these algorithms are inherently fine grained and experiments show that PRAM algorithms do often not achieve the expected speedup on real machines because of large message overheads. In this paper, we present coarse grained parallel graph algorithms with small message overheads that solve the following standard graph problems related to graph matching: finding maximum matchings in convex bipartite graphs, and finding maximum weight matchings in trees. To our knowledge, these are the first efficient parallel algorithms for these problems that are designed for standard commercial parallel machines such as off-the-shelf processor clusters.

Perfect one-factorizations in line-graphs and planar graphs.

Abstract: A one-factorization of a regular graph G is perfect if the union of any two one-factors is a Hamiltonian cycle. A graph G is said to be P1F if it possess a perfect one-factorization. We prove that G is a P1F cubic graph if and only if L(G) is a P1F quartic graph. Moreover, we give some necessary conditions for the existence of a P1F planar graph.

On Potentially 3-regular graph graphic Sequences

Abstract: For given a graph $H$, a graphic sequence $\pi=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $\pi$ containing $H$ as a subgraph. In this paper, we characterize the potentially $H$-graphic sequences where $H$ denotes 3-regular graph with 6 vertices. In other words, we characterize the potentially $K_{3,3}$ and $K_6-C_6$-graphic sequences where $K_{r,r}$ is an $r\times r$ complete bipartite graph. One of these characterizations implies a theorem due to Yin [25].

Hamiltonian cycle in almost distance-hereditary graphs with degree condition restricted to claws†

Abstract: Let G = (V(G), E(G)) be a connected graph. The distance between two vertices x and y in G, denoted by dG (x, y), is the length of a shortest path between x and y. A graph G is called almost distance-hereditary, if each connected induced subgraph H of G has the property that dH (u, v) ≤ dG (u, v) + 1 for every pair of vertices u and v in H. A graph G is 2-heavy if the degree of at least two end vertices of each claw in G are greater than or equal to |V(G)|/2. We show that every 2-connected, 2-heavy and almost distance-hereditary graph has a Hamiltonian cycle. This generalizes the main result of [8] that states: every 2-connected, claw-free and almost distance-hereditary graph is Hamiltonian. †Dedicated to Prof. Dr Dr h.c. Hubertus Th. Jongen on his 60th birthday.

Graph Algorithms for Improving Type-Logical Proof Search

Abstract: Proof nets are a graph theoretical representation of proofs in various fragments of type-logical grammar. In spite of this basis in graph theory, there has been relatively little attention to the use of graph theoretic algorithms for type-logical proof search. In this paper we will look at several ways in which standard graph theoretic algorithms can be used to restrict the search space. In particular, we will provide an O(n4) algorithm for selecting an optimal axiom link at any stage in the proof search as well as a O(kn3) algorithm for selecting the k best proof candidates.

Hamiltonian Strongly Regular Graphs

Abstract: We give a sufficient condition for a distance-regular graph to be Hamiltonian. In particular, the Petersen graph is the only connected non-Hamiltonian strongly regular graph on fewer than 99 vertices.

An Algorithm of System Decomposition Based on Laplace Spectral Graph Partitioning Technology

Abstract: In this paper we propose a spectral-graph-partitioning-based algorithm to decompose an object-oriented system into components. We begin with a weighted class dependence graph, in which vertexes stand for the classes and edges stand for the weight of the relationship between classes. We employ a technology from algebraic graph theory known as Laplace spectral graph partitioning to divide the class graph into components. The decomposition algorithm can be performed automatically and achieve a good performance.

Determination of Hamiltonian Circuit and Euler Trail from a Given Graphic Degree Sequence

Abstract: A sequence of nonnegative integers can represent degrees of a graph. Already there is an established condition under which the above conclusion is true. It is our objective in this paper to show that a similar degree sequence represents degrees of vertices of a graph having Hamiltonian Circuit or Euler Tour. The only fallacy of the conclusion is that there may be graphs those are not Hamiltonian or Euler graphs may have the same degree sequence.

Visualization of Graph based on the Three-dimensional Spring Model

Abstract: A three-dimensional (3D) spring modeling algorithm is given for the 3D visualization of graph, which have been implemented in OpenGL. It avoids edge crossings more effectively, and strengthens the sense of tridimensionality and aesthetics of graph while laying out the graph in three dimensions. By the use of interactive computer visualization, we can observe the distributing character of undirected graph from multi-angles.

Random Graph Model Simulations of Semantic Networks for Associative Concept Dictionaries

Abstract: Word association data in dictionary form can be simulated through the combination of three components: a bipartite graph with an imbalance in set sizes; a scale-free graph of the Barabasi-Albert model; and a normal distribution connecting the two graphs. Such a model makes it possible to simulate the complex features in degree distributions and the interesting graph clustering results that are typically observed for real data.

Degree Condition for Subdivisions of Unicyclic Graphs

Abstract: If H is any graph of order n with k non-trivial components, each of which contains at most one cycle, then every graph of order at least n and minimum degree at least n − k contains a subdivision of H such that only edges contained in a cycle in H are subdivided.

Graph weights arising from Mayer and Ree-Hoover theories of virial expansions

Abstract: We study graph weights (i.e., graph invariants) which arise naturally in Mayer's theory and Ree-Hoover's theory of virial expansions in the context of a non-ideal gas. We give special attention to the Second Mayer weight $w_M(c)$ and the Ree-Hoover weight $w_{RH}(c)$ of a $2$-connected graph $c$ which arise from the hard-core continuum gas in one dimension. These weights are computed using signed volumes of convex polytopes naturally associated with the graph $c$. Among our results are the values of Mayer's weight and Ree-Hoover's weight for all $2$-connected graphs $b$ of size at most $8$, and explicit formulas for certain infinite families.