Showing papers on "Quartic graph published in 2017"

Autoregressive Moving Average Graph Filtering

Abstract: One of the cornerstones of the field of signal processing on graphs are graph filters, direct analogs of classical filters, but intended for signals defined on graphs. This paper brings forth new insights on the distributed graph filtering problem. We design a family of autoregressive moving average (ARMA) recursions, which are able to approximate any desired graph frequency response, and give exact solutions for specific graph signal denoising and interpolation problems. The philosophy to design the ARMA coefficients independently from the underlying graph renders the ARMA graph filters suitable in static and, particularly, time-varying settings. The latter occur when the graph signal and/or graph topology are changing over time. We show that in case of a time-varying graph signal, our approach extends naturally to a two-dimensional filter, operating concurrently in the graph and regular time domain. We also derive the graph filter behavior, as well as sufficient conditions for filter stability when the graph and signal are time varying. The analytical and numerical results presented in this paper illustrate that ARMA graph filters are practically appealing for static and time-varying settings, as predicted by theoretical derivations.

$\mathsf {pSCAN}$ : Fast and Exact Structural Graph Clustering

Abstract: We study the problem of structural graph clustering, a fundamental problem in managing and analyzing graph data. Given an undirected unweighted graph, structural graph clustering is to assign vertices to clusters, and to identify the sets of hub vertices and outlier vertices as well, such that vertices in the same cluster are densely connected to each other while vertices in different clusters are loosely connected. In this paper, we develop a new two-step paradigm for scalable structural graph clustering based on our three observations. Then, we present a $\mathsf {pSCAN}$ approach, within the paradigm, aiming to reduce the number of structural similarity computations, and propose optimization techniques to speed up checking whether two vertices are structure-similar. $\mathsf {pSCAN}$ outputs exactly the same clusters as the existing approaches $\mathsf {SCAN}$ and $\mathsf {SCAN\text{++}}$ , and we prove that $\mathsf {pSCAN}$ is worst-case optimal. Moreover, we propose efficient techniques for updating the clusters when the input graph dynamically changes, and we also extend our techniques to other similarity measures, e.g., Jaccard similarity. Performance studies on large real and synthetic graphs demonstrate the efficiency of our new approach and our dynamic cluster maintenance techniques. Noticeably, for the twitter graph with 1 billion edges, our approach takes 25 minutes while the state-of-the-art approach cannot finish even after 24 hours.

Graph Classification via Deep Learning with Virtual Nodes.

Abstract: Learning representation for graph classification turns a variable-size graph into a fixed-size vector (or matrix). Such a representation works nicely with algebraic manipulations. Here we introduce a simple method to augment an attributed graph with a virtual node that is bidirectionally connected to all existing nodes. The virtual node represents the latent aspects of the graph, which are not immediately available from the attributes and local connectivity structures. The expanded graph is then put through any node representation method. The representation of the virtual node is then the representation of the entire graph. In this paper, we use the recently introduced Column Network for the expanded graph, resulting in a new end-to-end graph classification model dubbed Virtual Column Network (VCN). The model is validated on two tasks: (i) predicting bio-activity of chemical compounds, and (ii) finding software vulnerability from source code. Results demonstrate that VCN is competitive against well-established rivals.

On the calculation of resonances by analytic continuation of eigenvalues from the stabilization graph.

Abstract: Resonances play a major role in a large variety of fields in physics and chemistry. Accordingly, there is a growing interest in methods designed to calculate them. Recently, Landau et al. proposed a new approach to analytically dilate a single eigenvalue from the stabilization graph into the complex plane. This approach, termed Resonances Via Pade (RVP), utilizes the Pade approximant and is based on a unique analysis of the stabilization graph. Yet, analytic continuation of eigenvalues from the stabilization graph into the complex plane is not a new idea. In 1975, Jordan suggested an analytic continuation method based on the branch point structure of the stabilization graph. The method was later modified by McCurdy and McNutt, and it is still being used today. We refer to this method as the Truncated Characteristic Polynomial (TCP) method. In this manuscript, we perform an in-depth comparison between the RVP and the TCP methods. We demonstrate that while both methods are important and complementary, the advantage of one method over the other is problem-dependent. Illustrative examples are provided in the manuscript.

On the Spectra of Commuting and Non Commuting Graph on Dihedral Group

Abstract: Study about spectra of graph has became interesting work as well as study about commuting and non commuting graph of a group or a ring But the study about spectra of commuting and non commuting graph of dihedral group has not been done yet In this paper, we investigate adjacency spectrum, Laplacian spectrum, signless Laplacian spectrum, and detour spectrum of commuting and non commuting graph of dihedral group D 2 n

Survey of Graph Energies

Abstract: Let graph energy is a graph--spectrum--based quantity‎, ‎introduced in the 1970s‎. ‎After a latent period of 20--30 years‎, ‎it became a popular topic of research both‎ ‎in mathematical chemistry and in ``pure'' spectral graph theory‎, ‎resulting in‎ ‎over 600 published papers‎. ‎Eventually‎, ‎scores of different graph energies have‎ ‎been conceived‎. ‎In this article we provide the basic facts on graph energies‎, ‎in particular historical and bibliographic data.‎

A Partial Inverse Problem for the Differential Pencil on a Star-Shaped Graph

Abstract: The quadratic pencil of Sturm–Liouville operators on a star-shaped graph is investigated. We suppose that the coefficients of the pencil are known for all the edges of the graph except one, and study the partial inverse problem, which consists in recovering the remaining coefficients from the part of the spectrum. We prove the uniqueness theorem and provide a constructive algorithm for the solution of the partial inverse problem.

Graph Transform Learning for Image Compression

Abstract: In this paper, we propose a new graph-based coding framework and illustrate its application to image compression. Our approach relies on the careful design of a graph that optimizes the overall rate-distortion performance through an effective graph-based transform. We introduce a novel graph estimation algorithm, which uncovers the connectivities between the graph signal values by taking into consideration the coding of both the signal and the graph topology in rate-distortion terms. In particular, we introduce a novel coding solution for the graph by treating the edge weights as another graph signal that lies on the dual graph. Then, the cost of the graph description is introduced in the optimization problem by minimizing the sparsity of the coefficients of its graph Fourier transform (GFT) on the dual graph. In this way, we obtain a convex optimization problem whose solution defines an efficient transform coding strategy. The proposed technique is a general framework that can be applied to different types of signals, and we show two possible application fields, namely natural image coding and piecewise smooth image coding. The experimental results show that the proposed method outperforms classical fixed transforms such as DCT, and, in the case of depth map coding, the obtained results are even comparable to the state-of-the-art graph-based coding method, that are specifically designed for depth map images.

Graph algebras and the Gelfand–Kirillov dimension

Abstract: We study some properties of the Gelfand–Kirillov dimension in a non-necessarily unital context, in particular, its Morita invariance when the algebras have local units, and its commutativity with d...

Proof of Komlós's conjecture on Hamiltonian subsets

Abstract: Komlos conjectured in 1981 that among all graphs with minimum degree at least d, the complete graph Kd+1 minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when d is sufficiently large. In fact we prove a stronger result: for large d, any graph G with average degree at least d contains almost twice as many Hamiltonian subsets as Kd+1, unless G is isomorphic to Kd+1 or a certain other graph which we specify.

Towards an Integrated Graph Algebra for Graph Pattern Matching with Gremlin

Abstract: Graph data management has revealed beneficial characteristics in terms of flexibility and scalability by differently balancing between query expressivity and schema flexibility. This has resulted into an rapid developing new task specific graph systems, query languages and data models, such as property graphs, key-value, wide column, resource description framework (RDF), etc. Present day graph query languages are focused towards flexible graph pattern matching (aka sub-graph matching), where as graph computing frameworks aim towards providing fast parallel (distributed) execution of instructions. The consequence of this rapid growth in the variety of graph based data management systems has resulted in a lack of standardization. Gremlin, a graph traversal language and machine, provides a common platform for supporting any graph computing system (such as an OLTP graph database or OLAP graph processors). We present a formalization of graph pattern matching for Gremlin queries. We also study, discuss and consolidate various existing graph algebra operators into an integrated graph algebra.

Quantum Symmetries of Graph C*-Algebras

Abstract: The study of graph C*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have never been computed so far. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph C*-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph C*-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

The first zagreb index and some hamiltonian properties of the line graph of a graph

Abstract: The first Zagreb index of a graph is defined as the sum of the squares of the degrees of the vertices in the graph. Using first Zagreb index of a graph, we in this note present sufficient conditions for some Hamiltonian properties of the line graph of a graph.

Joint Cuts and Matching of Partitions in One Graph

Abstract: As two fundamental problems, graph cuts and graph matching have been investigated over decades, resulting in vast literature in these two topics respectively. However the way of jointly applying and solving graph cuts and matching receives few attention. In this paper, we first formalize the problem of simultaneously cutting a graph into two partitions i.e. graph cuts and establishing their correspondence i.e. graph matching. Then we develop an optimization algorithm by updating matching and cutting alternatively, provided with theoretical analysis. The efficacy of our algorithm is verified on both synthetic dataset and real-world images containing similar regions or structures.

Sharp bounds for the general Randić index $R_-1$ of a graph

Abstract: Let $G$ be an undirected simple, connected graph with $n \geq 3$ vertices and $m$ edges, with vertex degree sequence $d_1\ge d_2 \ge \cdots \ge d_n$. The general Randi\'c index is defined by \[ R_{-1}=\sum _{(i,j)\in E}\frac {1}{d_id_j}. \] Lower and upper bounds for $R_{-1}$ are obtained in this paper.

An M-channel critically sampled filter bank for graph signals

Abstract: We investigate an M-channel critically sampled filter bank for graph signals where each of the M filters is supported on a different subband of the graph Laplacian spectrum. We partition the graph vertices such that the mth set comprises a uniqueness set for signals supported on the mth subband. For analysis, the graph signal is filtered on each subband and downsampled on the corresponding set of vertices. However, the classical synthesis filters are replaced with interpolation operators, circumventing the issue of how to design a downsampling pattern and graph spectral filters to ensure perfect reconstruction for signals that do not reside on bipartite graphs. The resulting transform is critically sampled and graph signals are perfectly reconstructable from their analysis coefficients. We empirically explore the joint vertex-frequency localization of the dictionary atoms and sparsity of the analysis coefficients, as well as the ability of the proposed transform to compress piecewise-smooth graph signals.

Semi-supervised Orthogonal Graph Embedding with Recursive Projections

Abstract: Many graph based semi-supervised dimensionality reduction algorithms utilize the projection matrix to linearly map the data matrix from the original feature space to a lower dimensional representation. But the dimensionality after reduction is inevitably restricted to the number of classes, and the learned non-orthogonal projection matrix usually fails to preserve distances well and balance the weight on different projection direction. This paper proposes a novel dimensionality reduction method, called the semi-supervised orthogonal graph embedding with recursive projections (SOGE). We integrate the manifold smoothness and label fitness as well as the penalization of the linear mapping mismatch, and learn the orthogonal projection on the Stiefel manifold that empirically demonstrates better performance. Moreover, we recursively update the projection matrix in its orthocomplemented space to continuously learn more projection vectors, so as to better control the dimension of reduction. Comprehensive experiment on several benchmarks demonstrates the significant improvement over the existing methods.

The Pullback-Pushout Approach to Algebraic Graph Transformation

Abstract: Some recent algebraic approaches to graph transformation include a pullback construction involving the match, that allows one to specify the cloning of items of the host graph. We pursue further this trend by proposing the Pullback-Pushout (pb-po) Approach, where we combine smoothly the classical modifications to a host graph specified by a rule (a span of graph morphisms) with the cloning of structures specified by another rule. The approach is shown to be a conservative extension of agree (and thus of the sqpo approach), and we show that it can be extended with standard techniques to attributed graphs. We discuss conditions to ensure a form of locality of transformations, and conditions to ensure that the attribution of transformed graphs is total.

Nonlinear standing waves on planar branched systems: shrinking into metric graph

Abstract: We treat the stationary nonlinear Schroodinger equation on two-dimensional branched domains, so-called fat graphs. The shrinking limit when the domain becomes one-dimensional metric graph is studied by using analytical estimate of the convergence of fat graph boundary conditions into those for metric graph. Detailed analysis of such convergence on the basis of numerical solution of stationary nonlinear Schrodinger equation on a fat graph is provided. Possibility for reproducing different metric graph boundary conditions studied in earlier works is shown. Practical applications of the proposed model for such problems as Bose-Einstein condensation in networks, branched optical media, DNA, conducting polymers and wave dynamics in branched capillary networks are discussed.

On dynamics of graph maps with zero topological entropy

Abstract: We explore the dynamics of graph maps with zero topological entropy. It is shown that a continuous map $f$ on a topological graph $G$ has zero topological entropy if and only if it is locally mean equicontinuous, that is the dynamics on each orbit closure is mean equicontinuous. As an application, we show that Sarnak's Mobius Disjointness Conjecture is true for graph maps with zero topological entropy. We also extend several results known in interval dynamics to graph maps. We show that a graph map has zero topological entropy if and only if there is no $3$-scrambled tuple if and only if the proximal relation is an equivalence relation; a graph map has no scrambled pairs if and only if it is null if and only if it is tame.

Graph Structures in Bipolar Neutrosophic Environment

Abstract: A bipolar single-valued neutrosophic (BSVN) graph structure is a generalization of a bipolar fuzzy graph. In this research paper, we present certain concepts of BSVN graph structures. We describe some operations on BSVN graph structures and elaborate on these with examples. Moreover, we investigate some related properties of these operations.

Automorphism group of the total graph over a matrix ring

Abstract: Let be a finite field with q elements, be the ring of all matrices over , Z(R) be the set of all zero-divisors of R, i.e. Z(R) consists of all singular matrices over . The total graph of , denoted by , is a graph with all elements of R as vertices, and two distinct vertices are adjacent if and only if . In this paper, we determine the automorphisms of the total graph for .

Semi-linear Elliptic Equations on Graph

Abstract: By the mountain-pass theorem of Ambrosette-Rabinowitz, we prove that for any p>2, there exists a positive solution to −∆u+hu= |u|u in Ω. Using the same method, we prove similar results for the p-Laplacian equations. This partly improves recent results of Grigor’yan-Lin-Yang. AMS Subject Classifications: 34B45, 35A15, 58E30 Chinese Library Classifications: O175.27

Closed Form Solutions of Combinatorial Graph Laplacian Estimation under Acyclic Topology Constraints

Abstract: How to obtain a graph from data samples is an important problem in graph signal processing. One way to formulate this graph learning problem is based on Gaussian maximum likelihood estimation, possibly under particular topology constraints. To solve this problem, we typically require iterative convex optimization solvers. In this paper, we show that when the target graph topology does not contain any cycle, then the solution has a closed form in terms of the empirical covariance matrix. This enables us to efficiently construct a tree graph from data, even if there is only a single data sample available. We also provide an error bound of the objective function when we use the same solution to approximate a cyclic graph. As an example, we consider an image denoising problem, in which for each input image we construct a graph based on the theoretical result. We then apply low-pass graph filters based on this graph. Experimental results show that the weights given by the graph learning solution lead to better denoising results than the bilateral weights under some conditions.

The $1$-Yamabe equation on graph

Abstract: We study the following $1$-Yamabe equation on a connected finite graph $$\Delta_1u+g\mathrm{Sgn}(u)=h|u|^{\alpha-1}\mathrm{Sgn}(u),$$ where $\Delta_1$ is the discrete $1$-Laplacian, $\alpha>1$ and $g, h>0$ are known. We show that the above $1$-Yamabe equation always has a nontrivial solution $u\geq0$, $u eq0$.

Enumeration on graph mosaics

Abstract: Since the Jones polynomial was discovered, the connection between knot theory and quantum physics has been of great interest. Lomonaco and Kauffman introduced the knot mosaic system to give a definition of the quantum knot system that is intended to represent an actual physical quantum system. Recently the authors developed an algorithm producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. As a sequel to this research program, we similarly define the (embedded) graph mosaic system by using 16 graph mosaic tiles, representing graph diagrams with vertices of valence 3 and 4. We extend the algorithm to produce the exact number of all graph mosaics. The magnified state matrix that is an extension of the state matrix is mainly used.

Graph surfaces on five-dimensional sub-Lorentzian structures

Abstract: Studying the space-like graph surfaces of codimension 2 on the five-dimensional sub-Lorentzian structures with two negative directions of distinct degrees, we determine the differential properties of graph mappings and prove the area formula for the corresponding image surfaces.