Pair Of Graphs Research Articles

Multiplicatively Exact Algorithms for Transformation and Reconstruction of Directed Path-Cycle Graphs with Repeated Edges

For any weighted directed path-cycle graphs, a and b (referred to as structures), and any equal costs of operations (intermergings and duplication), we obtain an algorithm which, by successively applying these operations to a, outputs b if the first structure contains no paralogs (i.e., edges with a repeated name) and the second has no more than two paralogs for each edge. In finding the shortest sequence of operations to be applied to pass from a to b, the algorithm has a multiplicative error of at most 13/9 + ε, where ε is any strictly positive number, and its runtime is of the order of nO(ε−2.6), where n is the size of the input pair of graphs. In the case of no paralogs, equal sets of names in the structures, and equal operation costs, we have considered the following conditions on the transformation of a into b: all structures in them are from one cycle; all structures are from one path; all structures are from paths. For each of the conditions, we have obtained an exact (i.e., zero-error) quadratic time algorithm for finding the shortest transformation of a into b. For another list of operations (join and cut of a vertex, and deletion and insertion of an edge) over structures and for arbitrary costs of these operations, we have obtained an algorithm for the extension of structures specified at the leaves of a tree onto its interior vertices. The algorithm is exact if the tree is a star—in this case, structures in the leaves may even have unequal sets of names or paralogs. The runtime of the algorithm is of the order of nΧ + n2log(n), where n is the number of names in the leaves, and Χ is an easily computable characteristic of the structures in the leaves. In the general case, a cubic time algorithm finds a locally minimal solution.

The diagrammatic coaction beyond one loop

The diagrammatic coaction maps any given Feynman graph into pairs of graphs and cut graphs such that, conjecturally, when these graphs are replaced by the corresponding Feynman integrals one obtains a coaction on the respective functions. The coaction on the functions is constructed by pairing a basis of differential forms, corresponding to master integrals, with a basis of integration contours, corresponding to independent cut integrals. At one loop, a general diagrammatic coaction was established using dimensional regularisation, which may be realised in terms of a global coaction on hypergeometric functions, or equivalently, order by order in the ϵ expansion, via a local coaction on multiple polylogarithms. The present paper takes the first steps in generalising the diagrammatic coaction beyond one loop. We first establish general properties that govern the diagrammatic coaction at any loop order. We then focus on examples of two-loop topologies for which all integrals expand into polylogarithms. In each case we determine bases of master integrals and cuts in terms of hypergeometric functions, and then use the global coaction to establish the diagrammatic coaction of all master integrals in the topology. The diagrammatic coaction encodes the complete set of discontinuities of Feynman integrals, as well as the differential equations they satisfy, providing a general tool to understand their physical and mathematical properties.

Seurat games on Stockmeyer graphs

AbstractWe define a family of vertex colouring games played over a pair of graphs or digraphs by players and . These games arise from work on a longstanding open problem in algebraic logic. It is conjectured that there is a natural number such that always has a winning strategy in the game with colours whenever . This is related to the reconstruction conjecture for graphs and the degree‐associated reconstruction conjecture for digraphs. We show that the reconstruction conjecture implies our game conjecture with for graphs, and the same is true for the degree‐associated reconstruction conjecture and our conjecture for digraphs. We show (for any ) that the 2‐colour game can distinguish certain nonisomorphic pairs of graphs that cannot be distinguished by the ‐dimensional Weisfeiler–Leman algorithm. We also show that the 2‐colour game can distinguish the nonisomorphic pairs of graphs in the families defined by Stockmeyer as counterexamples to the original digraph reconstruction conjecture.

On a pair of fuzzy dominated mappings on closed ball in the multiplicative metric space with applications

The purpose of this paper is to establish some fixed point results for a pair of fuzzy dominated mappings satisfying contractive conditions on closed ball in multiplicative metric space. Some new fixed point results with graphic contraction on closed ball for a pair of fuzzy graph dominated mappings on multiplicative metric space have been established. Furthermore, we find a unique common solution for a system of non linear Voltera type integral equations and lastly we give an application to ensure the existence of common bounded solution of a functional equation in dynamic programming.

Graph similarity rectification for person search

In person search task, it is hard to retrieve the query persons undergoing large visual changes. To tackle this problem, we propose to exploit the context information to rectify the original individual similarity for better retrieval. To this end, we propose to model a query frame and a gallery frame as a graph pair, and then design the Siamese Residual Graph Convolutional Networks (SR-GCN) to aggregate context information to generate graph similarity as a complement of the original similarity. To model the relationships between context persons, we define the joint similarity adjacency matrix which assigns the proposed joint similarity as the edge weight to measure the contributions a context person makes to the aggregation. Therefore, the context node with a higher possibility to be a co-traveler of the target node makes more contributions to the matching of the target node. To further enhance the discriminative power of individual features, we also design a Random Proxy Center loss which explicitly constrains the intra-class variations to be smaller than the inter-class variations in the feature space and could make use of unlabeled samples. Experimental results on two public datasets show that our approach performs favorably against the state-of-the-art methods.

A large family of cospectral Cayley graphs over dicyclic groups

For a finite group G and an inverse closed subset S⊆G∖{e}, the Cayley graph X(G,S) has vertex set G and two vertices x,y∈G are adjacent if and only if xy−1∈S. Two graphs are called cospectral if their adjacency matrices have the same spectrum. Let p≥3 be a prime number and T4p be the dicyclic group of order 4p. In this paper, with the help of the characters from representation theory, we construct a large family of pairwise non-isomorphic and cospectral Cayley graphs over the dicyclic group T4p with p≥23, and find several pairs of non-isomorphic and cospectral Cayley graphs for 5≤p≤19.

MoCL: Data-driven Molecular Fingerprint via Knowledge-aware Contrastive Learning from Molecular Graph.

Recent years have seen a rapid growth of utilizing graph neural networks (GNNs) in the biomedical domain for tackling drug-related problems. However, like any other deep architectures, GNNs are data hungry. While requiring labels in real world is often expensive, pretraining GNNs in an unsupervised manner has been actively explored. Among them, graph contrastive learning, by maximizing the mutual information between paired graph augmentations, has been shown to be effective on various downstream tasks. However, the current graph contrastive learning framework has two limitations. First, the augmentations are designed for general graphs and thus may not be suitable or powerful enough for certain domains. Second, the contrastive scheme only learns representations that are invariant to local perturbations and thus does not consider the global structure of the dataset, which may also be useful for downstream tasks. In this paper, we study graph contrastive learning designed specifically for the biomedical domain, where molecular graphs are present. We propose a novel framework called MoCL, which utilizes domain knowledge at both local- and global-level to assist representation learning. The local-level domain knowledge guides the augmentation process such that variation is introduced without changing graph semantics. The global-level knowledge encodes the similarity information between graphs in the entire dataset and helps to learn representations with richer semantics. The entire model is learned through a double contrast objective. We evaluate MoCL on various molecular datasets under both linear and semi-supervised settings and results show that MoCL achieves state-of-the-art performance.

Edge counts for the auxiliary pair graph within the graphical unitary group approach

Closed-form expressions are presented for the numbers of edges in the auxiliary pair graphs (APGs) associated with non spin–orbit and spin–orbit Shavitt graphs for full configuration interaction expansions. A Shavitt graph is a visual representation of a configuration state function expansion space constructed via the graphical unitary group approach (GUGA). An APG is an organisational aid and a programmatic tool generated from a Shavitt graph. The number of edges in an APG determines bounds on the computational scaling as a function of the total numbers of electrons, orbitals, and spin multiplicities. The edge counts extend a suite of Shavitt graph statistics based on these functional parameters. The derivation and the presentation of the formulas for the edge counts has been assisted by the bra–ket interchange symmetry and the particle-hole interchange symmetry in the GUGA formalism. These symmetry operators produce one-to-one correspondences between various sets of edges, and this yields identities among some edge count formulas. There are 208 possible edge types. Of these, some do not contribute to two-electron operators, some are related by bra–ket interchange symmetry, and some are related by particle-hole interchange symmetry. For the remaining unique edge types, explicit expressions are derived for the numbers of edges.

Spectral conditions for graph rigidity in the Euclidean plane

Rigidity is the property of a structure that does not flex. It is well studied in discrete geometry and mechanics, and has applications in material science, engineering and biological sciences. A bar-and-joint framework is a pair (G,p) of graph G together with a map p of the vertices of G into a Euclidean space. We view the edges of (G,p) as bars and the vertices as universal joints. The vertices can move continuously as long as the distances between pairs of adjacent vertices are preserved. The framework is rigid if any such motion preserves the distances between all pairs of vertices. In 1970, Laman obtained a combinatorial characterization of rigid graphs in the Euclidean plane. In 1982, Lovász and Yemini discovered a new characterization and proved that every 6-connected graph is rigid. Combined with a combinatorial characterization of global rigidity given by Jackson and Jordán in 2009, it is actually proved that every 6-connected graph is globally rigid. Consequently, if the algebraic connectivity of a graph is greater than 5, then it is globally rigid. In this paper, we improve this bound and show that for a graph G with minimum degree δ≥6, if its algebraic connectivity is greater than 2+1δ−1, then G is rigid and if its algebraic connectivity is greater than 2+2δ−1, then G is globally rigid. Our results imply that every connected regular Ramanujan graph with degree at least 8 is globally rigid. We also prove a more general result giving a sufficient spectral condition for the existence of k edge-disjoint spanning rigid subgraphs. The same condition implies that a graph contains k edge-disjoint spanning 2-connected subgraphs. This result extends previous spectral conditions for packing edge-disjoint spanning trees.

GLMNet: Graph learning-matching convolutional networks for feature matching

Recently, graph convolutional networks (GCNs) have been employed for graph matching problem. It can integrate graph node feature embedding, node-wise affinity learning and matching optimization together in a unified end-to-end model. However, first, the matching graphs feeding to existing graph matching networks are generally fixed and independent of graph matching task, which thus are not guaranteed to be optimal for the graph matching task. Second, existing methods generally employ smoothing-based graph convolution to generate graph node embeddings, in which extensive smoothing convolution operation may dilute the desired discriminatory information of graph nodes. To overcome these issues, we propose a novel Graph Learning-Matching Network (GLMNet) for graph matching problem. GLMNet has three main aspects. (1) It integrates graph learning into graph matching which thus adaptively learns a pair of optimal graphs for graph matching task. (2) It further employs a Laplacian sharpening graph convolution to generate more discriminative node embeddings for graph matching. (3) A new constraint regularized loss is designed for GLMNet training which can encode the desired one-to-one matching constraints in matching optimization. Experiments demonstrate the effectiveness of GLMNet.

Graph convolutional networks for enhanced resolution 3D Electrical Capacitance Tomography image reconstruction

Three dimensional Electrical Capacitance Tomography (3D ECT) is an inexpensive tool for diagnosing non-conductive components of industrial processes. Although relatively mature, it still requires much work to improve its inverse nature of imaging capability. In particular, high resolution 3D ECT image reconstruction is very time-consuming and computationally heavy, and the best-known 3D ECT image reconstruction techniques have already reached their limits. Thus, there is a strong need to change a direction towards modern computational intelligence solutions. Therefore, this work proposes using graph convolutional networks (GCN) to raise the 3D ECT image quality. Mainly, it takes advantage of GCN’s ability to effectively use specific geometrical relationships hidden in the finite modeling unstructured grids commonly used to build 3D ECT images. These relationships are first encoded by a graph representing an ECT volumetric finite element grid. A GCN is next trained in a graph-to-graph framework with pairs of graphs representing high-quality nonlinear image reconstruction results as input and a simulated phantom as output. As a result, a trained GCN model fed with lower resolution 3D ECT image enhances its quality and spatial resolution. Tomographic image quality and resolution enhancement was evaluated using normalized mean square error and Pearson correlation coefficient, which improved by 35.5% and 3.74%, respectively.

Fast and efficient Rmap assembly using the Bi-labelled de Bruijn graph

Genome wide optical maps are high resolution restriction maps that give a unique numeric representation to a genome. They are produced by assembling hundreds of thousands of single molecule optical maps, which are called Rmaps. Unfortunately, there are very few choices for assembling Rmap data. There exists only one publicly-available non-proprietary method for assembly and one proprietary software that is available via an executable. Furthermore, the publicly-available method, by Valouev et al. (Proc Natl Acad Sci USA 103(43):15770–15775, 2006), follows the overlap-layout-consensus (OLC) paradigm, and therefore, is unable to scale for relatively large genomes. The algorithm behind the proprietary method, Bionano Genomics’ Solve, is largely unknown. In this paper, we extend the definition of bi-labels in the paired de Bruijn graph to the context of optical mapping data, and present the first de Bruijn graph based method for Rmap assembly. We implement our approach, which we refer to as rmapper, and compare its performance against the assembler of Valouev et al. (Proc Natl Acad Sci USA 103(43):15770–15775, 2006) and Solve by Bionano Genomics on data from three genomes: E. coli, human, and climbing perch fish (Anabas Testudineus). Our method was able to successfully run on all three genomes. The method of Valouev et al. (Proc Natl Acad Sci USA 103(43):15770–15775, 2006) only successfully ran on E. coli. Moreover, on the human genome rmapper was at least 130 times faster than Bionano Solve, used five times less memory and produced the highest genome fraction with zero mis-assemblies. Our software, rmapper is written in C++ and is publicly available under GNU General Public License at https://github.com/kingufl/Rmapper.

Simultaneous visibility representations of undirected pairs of graphs

We consider the problem of determining if a pair of undirected graphs 〈Gv,Gh〉, which share the same vertex set, has a representation using opaque geometric shapes for vertices, and vertical (respectively, horizontal) visibility between shapes to determine the edges of Gv (respectively, Gh). While such a simultaneous visibility representation of two graphs can be determined efficiently if the direction of the required visibility for each edge is provided (and the vertex shapes are sufficiently simple), it was unclear if edge direction is critical for efficiency. Here, an edge directed from u to v implies that the shape representing u is below (respectively, left of) the shape for v in Gv (respectively, Gh). We show that the problem is NP-complete without that information, even for graphs that are only slightly more complex than paths. In addition, we characterize which pairs of paths have simultaneous visibility representations using fixed orientation L-shapes. This narrows the range of possible graph families for which determining simultaneous visibility representation is non-trivial yet not NP-hard.

Finite automata approach to reconstructibility of switched Boolean control networks

This paper investigates the reconstructibility of switched Boolean control networks (SBCNs). Several new types of definition about reconstructibility are proposed where the existence and arbitrariness of the switching signal and the input sequence are inconsistent. A weighted pair graph describing all pairs of states which are indistinguished is defined. A deterministic finite automata method is applied into the reconstructibility analysis. Accordingly, several algorithms are designed to determine these types of reconstructibility. Moreover, for a given reconstructible SBCN, an algorithm for determining the current state is provided. Finally, an example is provided to demonstrate the effectiveness of the proposed algorithms and results.

A generalization of Fiedler's lemma and the spectra of H-join of graphs

A new generalization of Fiedler's lemma is obtained by introducing the concept of the main function of a matrix. As applications, the universal spectra of the H-join of any graphs (possibly non-regular) and the adjacency spectra of the H-generalized join constrained by (arbitrary) vertex subsets are obtained. The adjacency spectra of the generalized corona of graphs is deduced from the spectra of the H-join of graphs. Also, the construction of infinitely many pairs of non-regular universal cospectral graphs is provided.

Duality of Graded Graphs Through Operads

Pairs of graded graphs, together with the Fomin property of graded graph duality, are rich combinatorial structures providing among other a framework for enumeration. The prototypical example is the one of the Young graded graph of integer partitions, allowing us to connect number of standard Young tableaux and numbers of permutations. Here, we use operads, that algebraic devices abstracting the notion of composition of combinatorial objects, to build pairs of graded graphs. For this, we first construct a pair of graded graphs where vertices are syntax trees, the elements of free nonsymmetric operads. This pair of graphs is dual for a new notion of duality called $\phi$-diagonal duality, similar to the ones introduced by Fomin. We also provide a general way to build pairs of graded graphs from operads, wherein underlying posets are analogous to the Young lattice. Some examples of operads leading to new pairs of graded graphs involving integer compositions, Motzkin paths, and $m$-trees are considered.

Shannon Capacity and the Categorical Product

Shannon OR-capacity $C_{\rm OR}(G)$ of a graph $G$, that is the traditionally more often used Shannon AND-capacity of the complementary graph, is a homomorphism monotone graph parameter therefore $C_{\rm OR}(F\times G)\leqslant\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$ holds for every pair of graphs, where $F\times G$ is the categorical product of graphs $F$ and $G$. Here we initiate the study of the question when could we expect equality in this inequality. Using a strong recent result of Zuiddam, we show that if this "Hedetniemi-type" equality is not satisfied for some pair of graphs then the analogous equality is also not satisfied for this graph pair by some other graph invariant that has a much "nicer" behavior concerning some different graph operations. In particular, unlike Shannon OR-capacity or the chromatic number, this other invariant is both multiplicative under the OR-product and additive under the join operation, while it is also nondecreasing along graph homomorphisms. We also present a natural lower bound on $C_{\rm OR}(F\times G)$ and elaborate on the question of how to find graph pairs for which it is known to be strictly less than the upper bound $\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$. We present such graph pairs using the properties of Paley graphs.

Maximum Likelihood Estimation and Graph Matching in Errorfully Observed Networks

Given a pair of graphs with the same number of vertices, the inexact graph matching problem consists in finding a correspondence between the vertices of these graphs that minimizes the total number of induced edge disagreements. We study this problem from a statistical framework in which one of the graphs is an errorfully observed copy of the other. We introduce a corrupting channel model, and show that in this model framework, the solution to the graph matching problem is a maximum likelihood estimator (MLE). Necessary and sufficient conditions for consistency of this MLE are presented, as well as a relaxed notion of consistency in which a negligible fraction of the vertices need not be matched correctly. The results are used to study matchability in several families of random graphs, including edge independent models, random regular graphs, and small-world networks. We also use these results to introduce measures of matching feasibility, and experimentally validate the results on simulated and real-world networks. Supplemental files for this article are available online.

Synchronous linear constraint system games

Mathematical models of quantum mechanics can be studied and distinguished using nonlocal games. We discuss a class of nonlocal games called synchronous linear constraint system (syncLCS) games. We unify two algebraic approaches to studying syncLCS games and relate these games to nonlocal games played on graphs, known as graph isomorphism games. In more detail, syncLCS games are nonlocal games that verify whether or not two players share a solution to a given system of equations. Two algebraic objects associated with these games encode information about the existence of perfect strategies. They are called the game algebra and the solution group. Here, we show that these objects are essentially the same, i.e., the game algebra is a suitable quotient of the group algebra of the solution group. We also demonstrate that syncLCS games are equivalent to graph isomorphism games on a pair of graphs parameterized by the linear system.

On eigenvalues of the reciprocal distance signless Laplacian matrix of graphs

For a simple connected graph [Formula: see text], the reciprocal transmission [Formula: see text] of a vertex [Formula: see text] is defined as [Formula: see text] The reciprocal distance signless Laplacian (briefly RDSL) matrix of a connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the Harary matrix (reciprocal distance matrix) of [Formula: see text] and [Formula: see text] is the diagonal matrix of the vertex reciprocal transmissions in [Formula: see text]. In this paper, we first investigate the RDSL spectrum of the join and also joined union of graphs and we construct new pairs of equienergetic graphs by using the joined union of regular graphs. Then we obtain some bounds on the spectral radius of reciprocal distance signless Laplacian matrix of [Formula: see text], which is the largest eigenvalue of the matrix [Formula: see text]. We also characterize the extremal graphs.