Showing papers on "Quartic graph published in 2007"

Testing Expansion in Bounded Degree Graphs

Abstract: We consider the problem of testing graph expansion (either vertex or edge) in the bounded degree model [2]. We give a property tester that given a graph with degree bound d, an expansion bound �, and a parameter " > 0, accepts the graph with high probability if its expansion is more than �, and rejects it with high probability if it is "-far from a graph with expansion � 0 with degree bound d, where � 0 < � is a function of �. For edge expansion, we obtain � 0 = ( � 2 d ), and for vertex expansion, we obtain � 0 = ( � 2 d2 ). In either case, the algorithm runs in time

Spectra of graph neighborhoods and scattering

Abstract: Let $(G_\epsilon)_{\epsilon>0}$ be a family of '$\epsilon$-thin' Riemannian manifolds modeled on a finite metric graph $G$, for example, the $\epsilon$-neighborhood of an embedding of $G$ in some Euclidean space with straight edges. We study the asymptotic behavior of the spectrum of the Laplace-Beltrami operator on $G_\epsilon$ as $\epsilon\to 0$, for various boundary conditions. We obtain complete asymptotic expansions for the $k$th eigenvalue and the eigenfunctions, uniformly for $k\leq C\epsilon^{-1}$, in terms of scattering data on a non-compact limit space. We then use this to determine the quantum graph which is to be regarded as the limit object, in a spectral sense, of the family $(G_\epsilon)$. Our method is a direct construction of approximate eigenfunctions from the scattering and graph data, and use of a priori estimates to show that all eigenfunctions are obtained in this way.

Top-k subgraph matching query in a large graph

Abstract: Recently, due to its wide applications, subgraph search has attracted a lot of attention from database and data mining community. Sub-graph search is defined as follows: given a query graph Q, we report all data graphs containing Q in the database. However, there is little work about sub-graph search in a single large graph, which has been used in many applications, such as biological network and social network.In this paper, we address top-k sub-graph matching query problem, which is defined as follows: given a query graph Q, we locate top-k matchings of Q in a large data graph G according to a score function. The score function is defined as the sum of the pairwise similarity between a vertex in Q and its matching vertex in G. Specifically, we first design a balanced tree (that is G-Tree) to index the large data graph. Then, based on G-Tree, we propose an efficient query algorithm (that is Ranked Matching algorithm). Our extensive experiment results show that, due to efficiency of pruning strategy, given a query with up to 20 vertices, we can locate the top-100 matchings in less than 10 seconds in a large data graph with 100K vertices. Furthermore, our approach outperforms the alternative method by orders of magnitude.

A dual version of the ribbon graph decomposition of moduli space

Abstract: This note gives a construction of a dual version of the ribbon graph decomposition of the moduli spaces of Riemann surfaces.

The Laplacian spectral radius of a graph under perturbation

Abstract: In this paper, we investigate how the Laplacian spectral radius changes when one graph is transferred to another graph obtained from the original graph by adding some edges, or subdivision, or removing some edges from one vertex to another.

Hamilton cycles in prisms

Abstract: The prism over a graph G is the Cartesian product G s K2 of G with the complete graph K2. If G is hamiltonian, then GsK2 is also hamiltonian but the converse does not hold in general. Having a hamiltonian prism is shown to be an interesting relaxation of being hamiltonian. In this article, we examine classical problems on hamiltonicity of graphs in the context of having a hamiltonian prism. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 249–269, 2007 Sadly, the last author passed away in September 2003.

Mathematical Morphology in Any Color Space

Abstract: In this paper, a new graph-based ordering of color vectors is presented for mathematical morphology purposes An attractive propoerty of the proposed ordering is its color space independence A complete graph is defined over a filter window and its structure is analyzed to construct an ordering of color vectors by finding a Hamiltonian path in a two-step algorithm

Noncrossing Hamiltonian paths in geometric graphs

Abstract: A geometric graph is a graph embedded in the plane in such a way that vertices correspond to points in general position and edges correspond to segments connecting the appropriate points. A noncrossing Hamiltonian path in a geometric graph is a Hamiltonian path which does not contain any intersecting pair of edges. In the paper, we study a problem asked by Micha Perles: determine the largest number h(n) such that when we remove any set of h(n) edges from any complete geometric graph on n vertices, the resulting graph still has a noncrossing Hamiltonian path. We prove that h(n)>=(1/22)n. We also establish several results related to special classes of geometric graphs. Let h"1(n) denote the largest number such that when we remove edges of an arbitrary complete subgraph of size at most h"1(n) from a complete geometric graph on n vertices the resulting graph still has a noncrossing Hamiltonian path. We prove that 12n

Two particles on a star graph I

Abstract: We consider a two particle system on a star graph with $\delta$-function interaction. A class of eigensolutions is described which are constructed from appropriate one particle solutions, and hence are parametrised by two momenta. These solutions include a family of solutions with discontinuous derivative on the diagonal.

The spectrum of the averaging operator on a network (metric graph)

Abstract: A network is a countable, connected graph $X$ viewed as a one-complex, where each edge $[x,y]=[y,x]$ ($x,y\in X^0$, the vertex set) is a copy of the unit interval within the graph's one-skeleton $X^1$ and is assigned a positive conductance $\mathsf{c}(xy)$. A reference "Lebesgue" measure $X^1$ is built up by using Lebesgue measure with total mass $\mathsf{c}(xy)$ on each edge $xy$. There are three natural operators on $X$: the transition operator $P$ acting on functions on $X^0$ (the reversible Markov chain associated with $\mathsf{c}$), the averaging operator $A$ over spheres of radius~1 on $X^1$, and the Laplace operator $\Delta$ on $X^1$ (with Kirchhoff conditions weighted by $\mathsf{c}$ at the vertices). The relation between the $\ell^2$-spectrum of $P$ and the $H^2$-spectrum of~$\Delta$ was described by {Cattaneo} \cite{Cat}. In this note we describe the relation between the $\ell^2$-spectrum of $P$ and the $L^2$-spectrum of $A$.

Each maximal planar graph with exactly two separating triangles is Hamiltonian

Abstract: A classical result of Whitney states that each maximal planar graph without separating triangles is Hamiltonian, where a separating triangle is a triangle whose removal separates the graph. Chen [Any maximal planar graph with only one separating triangle is Hamiltonian J. Combin. Optim. 7 (2003) 79-86] proved that any maximal planar graph with only one separating triangle is still Hamiltonian. In this paper, it is shown that the conclusion of Whitney's Theorem still holds if there are exactly two separating triangles.

Pauli graph and finite projective lines/geometries

Abstract: The commutation relations between the generalized Pauli operators of N -qudits (i.e., Np-level quantum sys-tems), and the structure of their maximal sets of commuting bases, follow a nice graph theoretical/geometricalpattern. One may identify vertices/points with the operators so that edges/lines join commuting pairs of themto form the so-called Pauli graph P p N . As per two-qubits ( p =2,N = 2) all basic properties and partitioningsof this graph are embodied in the geometry of the symplectic generalized quadrangle of order two, W (2). Thestructure of the two-qutrit ( p =3,N = 2) graph is more involved; here it turns out more convenient to dealwith its dual in order to see all the parallels with the two-qubit case and its surmised relation with the geom-etry of generalized quadrangle Q (4 ,3), the dual of W (3). Finally, the generalized adjacency graph for multiple(N> 3) qubits/qutrits is shown to follow from symplectic polar spaces of order two/three. The relevance ofthese mathematical concepts to mutually unbiased base s and to quantum entanglement is also highlighted insome detail.Keywords: Generalized Pauli operators, Pauli graph, Quantum entanglement, Mutually unbiased bases, Gen-eralized quadrangles, Symplectic polar spaces

Separable Decomposition of Graph Using α-cliques

Abstract: The article deals with an evolutionary based method to decompose graph into strongly connected structures, we called α-cliques. The α-clique is a generalization of a clique concept with the introduction of parameter α. Using this parameter it is possible to control the degree (or strength) of connections among vertices (nodes) of this sub-graph structure. The evolutionary approach is proposed as a method that enables to find separate α-cliques that cover the set of graph vertices.

A note on zero divisor graph over rings

Abstract: In this article we discuss the graphs of the sets of zero-divisors of a ring Now let R be a ring Let G be a graph with elements of R as vertices such that two non-zero elements a, b ∈ R are adjacent if ab = ba = 0 We examine such a graph and try to find out when such a graph is planar and when is it complete etc Mathematics Subject Classification: Primary 16-xx, 05-xx; Secondary 05C50

Almost Hamiltonian Cubic Graphs

Abstract: A Hamiltonian walk in a connected graph G of order n is a closed spanning walk of minimum length in G. For a connected graph G, let h(G) be the length of a Hamiltonian walk in G and call it the Hamiltonian number of G. Let i be a non-negative integer. A connected graph G of order n is called an i-Hamiltonian if h(G) = n+i. Thus a 0-Hamiltonian graph is Hamiltonian. A 1-Hamiltonian graph is called an almost Hamiltonian graph. We prove in this paper that for an even integer n ≥ 10 there exists an almost Hamiltonian cubic graph of order n. Let P(k, m) be the generalized Petersen graph of order 2k. We show that P(k, m) is an almost Hamiltonian graph if and only if m= 2 and k ≡ 5(mod 6). For a cubic graph G, we define G* to be the graph obtained from G by replacing each vertex of G to a triangle, matching the vertices of the triangle to the former neighbors of the replaced vertex. We show that G is Hamiltonian if and only if G* is Hamiltonian and if G is almost Hamiltonian then G* is 2-Hamiltonian.

How Much Work Does It Take To Straighten a Plane Graph Out

Abstract: We prove that if one wants to make a plane graph drawing straight-line then in the worst case one has to move almost all vertices.

Closure for the property of having a hamiltonian prism

Abstract: We prove that a graph G of order n has a hamiltonian prism if and only if the graph Cl4n-3–4-3(G) has a hamiltonian prism where Cl4n-3–4-3(G) is the graph obtained from G by sequential adding edges between non-adjacent vertices whose degree sum is at least 4n-3–4-3. We show that this cannot be improved to less than 4n-3–5. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 209–220, 2007

Eigensolution for Stability Analysis of Planar Frames by Graph Symmetry

Abstract: : In this article, systematic methods are developed for calculating the buckling loads of symmetric planar frames The mechanical properties of elements of a structure are assigned to the members of the graph model, resulting in a weighted graph Factorization of this graph results in subgraphs for which the eigenvalues can easily be calculated Decomposition and healing process used in this article reduce the dimensions of the matrices required for stability analysis of symmetric frames Therefore, the accuracy of calculation increases and the cost of the computation decreases with a fixed discretization, in comparison with classic methods

On a class of nonlinear variational inequalities: High concentration of the graph of weak solution via its fractional dimension and Minkowski content

Abstract: Weak continuous bounded solutions of a class of nonlinear variational inequalities associated to one-dimensional $p$-Laplacian are studied. It is shown that a kind of boundary behaviour of nonlinearity in the main problem produces a kind of high boundary concentration of the graph of solutions. It is verified by calculating lower bounds for the upper Minkowski-Bouligand dimension and Minkowski content of the graph of each solution and its derivative. Finally, the order of growth for singular behaviour of the $L^{; ; ; ; p}; ; ; ; $ norm of derivative of solutions is given.

Hamiltonian cycles avoiding sets of edges in a graph

Abstract: A spanning cycle in a graph G is called a hamiltonian cycle, and if such a cycle exists G is said to be hamiltonian. Let G be a graph and H be a subgraph of G. If G contains a hamiltonian cycle C such that E(C) \ E(H) is empty, we say that C is an H-avoiding hamiltonian cycle. Let F be any graph. If G contains an H-avoiding hamiltonian cycle for every subgraph H of G such that H = F, then we say that G is F-avoiding hamiltonian. In this paper, we give minimum degree and degree-sum conditions which assure that a graph G is F-avoiding hamiltonian for various choices of F. In particular, we consider the cases where F is a union of k edge-disjoint hamiltonian cycles or a union of k edge-disjoint perfect matchings. If G is F-avoiding hamiltonian for any such F, then it is possible to extend families of these types in G. Finally, we undertake a discussion of F-avoiding pancyclic graphs.

World Graph Formalism for Feynman Amplitudes

Abstract: A unified treatment of Schwinger parametrised Feynman amplitudes is suggested which addresses vertices of arbitrary order on the same footing as propagators. Contributions from distinct diagrams are organised collectively. The scheme is based on the continuous graph Laplacian. The analogy to a classical statistical diffusion system of vector charges on the graph is explored.

Some graph properties determined by edge zeta functions

Abstract: Stark and Terras introduced the edge zeta function of a finite graph in 1996 The edge zeta function is the reciprocal of a polynomial in twice as many variables as edges in the graph and can be computed in polynomial time We look at graph properties which we can determine using the edge zeta function In particular, the edge zeta function is enough to deduce the clique number, the number of Hamiltonian cycles, and whether a graph is perfect or chordal Actually computing these properties takes exponential time Finally, we present a new example illustrating that the Ihara zeta function cannot necessarily do the same

Graph partitioning based on link distributions

Abstract: Existing graph partitioning approaches are mainly based on optimizing edge cuts and do not take the distribution of edge weights (link distribution) into consideration. In this paper, we propose a general model to partition graphs based on link distributions. This model formulates graph partitioning under a certain distribution assumption as approximating the graph affinity matrix under the corresponding distortion measure. Under this model, we derive a novel graph partitioning algorithm to approximate a graph affinity matrix under various Bregman divergences, which correspond to a large exponential family of distributions. We also establish the connections between edge cut objectives and the proposed model to provide a unified view to graph partitioning.

A parallel algorithm for interpolation in pancake graph

Abstract: In this paper a parallel algorithm for Lagrange interpolation is applied on a n-pancake graph The n- pancake graph is a Cayley graph with N=n! vertices and with attractive properties regarding degree, diameter, symmetry, embeddings and self similarity Using these properties the algorithm carries the calculation in O(N) steps of communication and arithmetic operations instead of O(N2) steps for a single processor system

On cubic 2-independent Hamiltonian connected graphs

Abstract: A graph G=(V,E) is Hamiltonian connected if there exists a Hamiltonian path between any two vertices in G. Let P1=(u1,u2,…,u|V|) and P2=(v1,v2,…,v|V|) be any two Hamiltonian paths of G. We say that P1 and P2 are independent if u1=v1,u|V|=v|V|, and ui≠vi for 1

Combining smooth graphs with semi-supervised learning

Abstract: The key points of the semi-supervised learning problem are the label smoothness and cluster assumptions. In graph-based semi-supervised learning, graph representations of the data are so important that different graph representations can affect the classification results heavily. We present a novel method to produce a graph called smooth Markov random walk graph which takes into account the two assumptions employed by semi-supervised learning. The new graph is achieved by modifying the eigenspectrum of the transition matrix of Markov random walk graph and is sufficiently smooth with respect to the intrinsic structure of labeled and unlabeled points.We believe the smoother graph will benefit semi-supervised learning. Experiments on artificial and real world dataset indicate that our method provides superior classification accuracy over several state-of-the-art methods.

Graph Algebra: Mathematical Modeling With a Systems Approach

Abstract: Series Editor's Introduction Acknowledgments 1. Systems Analysis Structure and Function An Overview of the Substantive Examples Found in Subsequent Chapters 2. Graph Algebra Basics Inputs, Outputs, and the Forward Path Feedback Loops and Mason's Rule An Example From Economics: The Keynesian Multiplier 3. Graph Algebra and Discrete-Time Linear Operators Delay and Advance Operators for Discrete Time Including an Additive Constant With Graph Algebra Difference and Summation Operators for Discrete Time An Estimated Example: Labor Union Membership 4. Working With Systems of Equations Richardson's Arms Race Model Using Graph Algebra Variations of Richardson's Arms Race Model An Estimated Example of a Multiple-Equation System With Nonlinear or Embedded Parameters: Richardson's Arms Race 5. Applying Graph Algebra to Continuous Time Using Graph Algebra With Continuous-Time Operators 6. Graph Algebra and Nonlinearity Nonlinear Filters The Logistic Function Placement Rules for Operators in Nonlinear Models Graph Algebra and Chaos Forced Oscillators 7. Working With Conditional Paths Logical and Decision Systems An Example of Democratic Transition 8. Systems, Shocks, and Stochasticity 9. Graph Algebra and Social Theory Systems and Equilibria References Index About the Author