Set Of Graphs Research Articles

Separation and relative quasiconvexity criteria for relatively geometric actions

Bowditch characterized relative hyperbolicity in terms of group actions on fine hyperbolic graphs with finitely many edge orbits and finite edge stabilizers. In this paper, we define generalized fine actions on hyperbolic graphs, in which the peripheral subgroups are allowed to stabilize finite subgraphs rather than stabilizing a point. Generalized fine actions are useful for studying groups that act relatively geometrically on a CAT(0) cube complex, which were recently defined by the first two authors. Specifically, we show that any group acting relatively geometrically on a CAT(0) cube complex admits a generalized fine action on the one-skeleton of the cube complex. For generalized fine actions, we prove a criterion for relative quasiconvexity of subgroups that cocompactly stabilize quasiconvex subgraphs, generalizing a result of Martínez-Pedroza and Wise in the setting of fine hyperbolic graphs. As an application, we obtain a characterization of boundary separation in generalized fine graphs and use it to prove that Bowditch boundary points in relatively geometric actions are always separated by a hyperplane stabilizer.

Lict k-Domination in Graphs

Objectives: In this study, we find lict -domination number of various types of graphs. Methods: Let be any graph, a set is said to be an -dominating set of lict graph if every vertex is dominated by at least vertices in , that is . The Lict -domination number is the minimum cardinality of -dominating set of . Findings: This study is centered on the lict -domination number of the graph and developed its relationship with other different domination parameters. Novelty: This study introduces the concept of “Lict -Domination in Graphs”. It obtains many bounds on in terms of vertices, edges, and other different parameters of . Keywords: Domination number, k-domination number, Lict graph, Lict k-domination number, Total domination number, Independent domination number

Dynamic Spiking Graph Neural Networks

The integration of Spiking Neural Networks (SNNs) and Graph Neural Networks (GNNs) is gradually attracting attention due to the low power consumption and high efficiency in processing the non-Euclidean data represented by graphs. However, as a common problem, dynamic graph representation learning faces challenges such as high complexity and large memory overheads. Current work often uses SNNs instead of Recurrent Neural Networks (RNNs) by using binary features instead of continuous ones for efficient training, which overlooks graph structure information and leads to the loss of details during propagation. Additionally, optimizing dynamic spiking models typically requires the propagation of information across time steps, which increases memory requirements. To address these challenges, we present a framework named Dynamic Spiking Graph Neural Networks (Dy-SIGN). To mitigate the information loss problem, Dy-SIGN propagates early-layer information directly to the last layer for information compensation. To accommodate the memory requirements, we apply the implicit differentiation on the equilibrium state, which does not rely on the exact reverse of the forward computation. While traditional implicit differentiation methods are usually used for static situations, Dy-SIGN extends it to the dynamic graph setting. Extensive experiments on three large-scale real-world dynamic graph datasets validate the effectiveness of Dy-SIGN on dynamic node classification tasks with lower computational costs.

Reduction for asynchronous Boolean networks: elimination of negatively autoregulated components

To simplify the analysis of Boolean networks, a reduction in the number of components is often considered. A popular reduction method consists in eliminating components that are not autoregulated, using variable substitution. In this work, we show how this method can be extended, for asynchronous dynamics of Boolean networks, to the elimination of vertices that have a negative autoregulation, and study the effects on the dynamics and interaction structure. For elimination of non-autoregulated variables, the preservation of attractors is in general guaranteed only for fixed points. Here we give sufficient conditions for the preservation of complex attractors. The removal of so called mediator nodes (i.e. vertices with indegree and outdegree one) is often considered, and frequently does not affect the attractor landscape. We clarify that this is not always the case, and in some situations even subtle changes in the interaction structure can lead to a different asymptotic behaviour. Finally, we use properties of the more general elimination method introduced here to give an alternative proof for a bound on the number of attractors of asynchronous Boolean networks in terms of the cardinality of positive feedback vertex sets of the interaction graph.

Counting Rules for Computing the Number of Independent Sets of a Grid Graph

The issue of counting independent sets of a graph, G, represented as i(G), is a significant challenge within combinatorial mathematics. This problem finds practical applications across various fields, including mathematics, computer science, physics, and chemistry. In chemistry, i(G) is recognized as the Merrifield–Simmons (M-S) index for molecular graphs, which is one of the most relevant topological indices related to the boiling point in chemical compounds. This article introduces an innovative algorithm designed for tallying independent sets within grid-like structures. The proposed algorithm is based on the ‘branch-and-bound’ technique and is applied to compute i(Gm,n) for a square grid formed by m rows and n columns. The proposed approach incorporates the widely recognized vertex reduction rule as the basis for splitting the current subgraph. The methodology involves breaking down the initial grid iteratively until outerplanar graphs are achieved, serving as the ’basic cases’ linked to the leaf nodes of the computation tree or when no neighborhood is incident to a minimum of five rectangular internal faces. The time complexity of the branch-and-bound algorithm speeds up the computation of i(Gm,n) compared to traditional methods, like the transfer matrix method. Furthermore, the scope of the proposed algorithm is more general than the algorithms focused on grids since it could be applied to process general mesh graphs.

Predicting COVID-19 Vaccination Uptake Using a Small and Interpretable Set of Judgment and Demographic Variables: Cross-Sectional Cognitive Science Study.

Despite COVID-19 vaccine mandates, many chose to forgo vaccination, raising questions about the psychology underlying how judgment affects these choices. Research shows that reward and aversion judgments are important for vaccination choice; however, no studies have integrated such cognitive science with machine learning to predict COVID-19 vaccine uptake. This study aims to determine the predictive power of a small but interpretable set of judgment variables using 3 machine learning algorithms to predict COVID-19 vaccine uptake and interpret what profile of judgment variables was important for prediction. We surveyed 3476 adults across the United States in December 2021. Participants answered demographic, COVID-19 vaccine uptake (ie, whether participants were fully vaccinated), and COVID-19 precaution questions. Participants also completed a picture-rating task using images from the International Affective Picture System. Images were rated on a Likert-type scale to calibrate the degree of liking and disliking. Ratings were computationally modeled using relative preference theory to produce a set of graphs for each participant (minimum R2>0.8). In total, 15 judgment features were extracted from these graphs, 2 being analogous to risk and loss aversion from behavioral economics. These judgment variables, along with demographics, were compared between those who were fully vaccinated and those who were not. In total, 3 machine learning approaches (random forest, balanced random forest [BRF], and logistic regression) were used to test how well judgment, demographic, and COVID-19 precaution variables predicted vaccine uptake. Mediation and moderation were implemented to assess statistical mechanisms underlying successful prediction. Age, income, marital status, employment status, ethnicity, educational level, and sex differed by vaccine uptake (Wilcoxon rank sum and chi-square P<.001). Most judgment variables also differed by vaccine uptake (Wilcoxon rank sum P<.05). A similar area under the receiver operating characteristic curve (AUROC) was achieved by the 3 machine learning frameworks, although random forest and logistic regression produced specificities between 30% and 38% (vs 74.2% for BRF), indicating a lower performance in predicting unvaccinated participants. BRF achieved high precision (87.8%) and AUROC (79%) with moderate to high accuracy (70.8%) and balanced recall (69.6%) and specificity (74.2%). It should be noted that, for BRF, the negative predictive value was <50% despite good specificity. For BRF and random forest, 63% to 75% of the feature importance came from the 15 judgment variables. Furthermore, age, income, and educational level mediated relationships between judgment variables and vaccine uptake. The findings demonstrate the underlying importance of judgment variables for vaccine choice and uptake, suggesting that vaccine education and messaging might target varying judgment profiles to improve uptake. These methods could also be used to aid vaccine rollouts and health care preparedness by providing location-specific details (eg, identifying areas that may experience low vaccination and high hospitalization).

Open problems on Sombor index of unicyclic and bicyclic graphs

Let Γ=(V,E) be a graph of order p. Recently, the Sombor index is introduced, defined asSO(Γ)=∑vivj∈E(Γ)dΓ(vi)2+dΓ(vj)2, where dΓ(vi) is the degree of the vertex vi in Γ. Cruz and Rada [4] obtained an upper bound on the Sombor index of unicyclic and bicyclic graphs of order p, but did not characterize the extremal graphs. In the same paper, they mentioned that the maximal graphs over the set of unicyclic and bicyclic graphs with respect to Sombor index, is an interesting problem that remains open. In this paper we completely solve these problems.

Inverse Domination Number and Inverse Total Domination of Sierpinski Star Graph

Given a graph G=(V(G),E(G)) consisting of the set of vertices V(G) and the set of edges E(G). For example, D(G) is a domination set of graph G with minimum cardinality, if V(G)-D(G) contains a domination set D^(-1) (G), then D^(-1) (G) is called the inverse domination set of graph G. The minimum cardinality of the inverse domination set of the graph G is called the inverse domination number, denoted by γ^(-1) (G). If D_t (G) is the total domination set of the graph G with minimal cardinality, and V(G)-D_t (G) contains the total domination set D_t^(-1) (G), then D_t^(-1) (G) is called the inverse total domination set of the graph G. The minimum cardinality of the inverse total domination set of the graph G is called the inverse total domination number, denoted by γ_t^(-1) (G). This paper discusses the inverse domination and the inverse total domination on the Sierpinski Star graph SS_n, obtained the inverse domination number γ^(-1) (SS_n )=0 for n&lt;3 and γ^(-1) (SS_n )=4∙3^(n-3) for n≥3 and the inverse total domination number γ_t^(-1) (SS_n )=0 for n≥1.

Max Weight Independent Set in Graphs with No Long Claws: An Analog of the Gyárfás’ Path Argument

We revisit recent developments for the Maximum Weight Independent Set problem in graphs excluding a subdivided claw S t,t,t as an induced subgraph and provide a subexponential-time algorithm with improved running time \(2^{\mathcal {O}(\sqrt {nt}\log n)}\) and a quasipolynomial-time approximation scheme with improved running time \(2^{\mathcal {O}(\varepsilon ^{-1}t \log ^{5} n)}\) . The Gyárfás’ path argument, a powerful tool that is the main building block for many algorithms in P t -free graphs, ensures that given an n -vertex P t -free graph, in polynomial time we can find a set P of at most t -1 vertices such that every connected component of G-N[P] has at most n /2 vertices. Our main technical contribution is an analog of this result for S t,t,t -free graphs: given an n -vertex S t,t,t -free graph, in polynomial time we can find a set P of \(\mathcal {O}(t \log n)\) vertices and an extended strip decomposition (an appropriate analog of the decomposition into connected components) of G-N[P] such that every particle (an appropriate analog of a connected component to recurse on) of the said extended strip decomposition has at most n /2 vertices.

Efficient High-Quality Clustering for Large Bipartite Graphs

A bipartite graph contains inter-set edges between two disjoint vertex sets, and is widely used to model real-world data, such as user-item purchase records, author-article publications, and biological interactions between drugs and proteins. k-Bipartite Graph Clustering (k-BGC) is to partition the target vertex set in a bipartite graph into k disjoint clusters. The clustering quality is important to the utility of k-BGC in various applications like social network analysis, recommendation systems, text mining, and bioinformatics, to name a few. Existing approaches to k-BGC either output clustering results with compromised quality due to inadequate exploitation of high-order information between vertices, or fail to handle sizable bipartite graphs with billions of edges. Motivated by this, this paper presents two efficient k-BGC solutions, HOPE and HOPE+, which achieve state-of-the-art performance on large-scale bipartite graphs. HOPE obtains high scalability and effectiveness through a new k-BGC problem formulation based on the novel notion of high-order perspective (HOP) vectors and an efficient technique for low-rank approximation of HOP vectors. HOPE+ further elevates the k-BGC performance to another level with a judicious problem transformation and a highly efficient two-stage optimization framework. Two variants, HOPE+ (FNEM) and HOPE+ (SNEM) are designed when either the Frobenius norm or spectral norm is applied in the transformation. Extensive experiments, comparing HOPE and HOPE+ against 13 competitors on 10 real-world datasets, exhibit that our solutions, especially HOPE+, are superior to existing methods in terms of result quality, while being up to orders of magnitude faster. On the largest dataset MAG with 1.1 billion edges, HOPE+ is able to produce clusters with the highest clustering accuracy within 31 minutes, which is unmatched by any existing solution for k-BGC.

Progressive Multiple Alignment of Graphs

The comparison of multiple (labeled) graphs with unrelated vertex sets is an important task in diverse areas of applications. Conceptually, it is often closely related to multiple sequence alignments since one aims to determine a correspondence, or more precisely, a multipartite matching between the vertex sets. There, the goal is to match vertices that are similar in terms of labels and local neighborhoods. Alignments of sequences and ordered forests, however, have a second aspect that does not seem to be considered for graph comparison, namely the idea that an alignment is a superobject from which the constituent input objects can be recovered faithfully as well-defined projections. Progressive alignment algorithms are based on the idea of computing multiple alignments as a pairwise alignment of the alignments of two disjoint subsets of the input objects. Our formal framework guarantees that alignments have compositional properties that make alignments of alignments well-defined. The various similarity-based graph matching constructions do not share this property and solve substantially different optimization problems. We demonstrate that optimal multiple graph alignments can be approximated well by means of progressive alignment schemes. The solution of the pairwise alignment problem is reduced formally to computing maximal common induced subgraphs. Similar to the ambiguities arising from consecutive indels, pairwise alignments of graph alignments require the consideration of ambiguous edges that may appear between alignment columns with complementary gap patterns. We report a simple reference implementation in Python/NetworkX intended to serve as starting point for further developments. The computational feasibility of our approach is demonstrated on test sets of small graphs that mimimc in particular applications to molecular graphs.

Spectral Radius of Graphs with Given Size and Odd Girth

Let $\mathcal{G}(m,k)$ be the set of graphs with size $m$ and odd girth (the length of shortest odd cycle) $k$. In this paper, we determine the graph maximizing the spectral radius among $\mathcal{G}(m,k)$ when $m$ is odd. As byproducts, we show that, there is a number $\eta(m,k)&gt;\sqrt{m-k+3}$ such that every non bipartite graph $G$ with size $m$ and spectral radius $\rho\ge \eta(m,k)$ must contain an odd cycle of length less than $k$ unless $m$ is odd and $G\cong SK_{k,m}$, which is the graph obtained by subdividing an edge $k-2$ times of the complete bipartite graph $K_{2,\frac{m-k+2}{2}}$. This result implies the main results of Zhai and Shu [Discrete Math. 345 (2022)] and settles a conjecture of Li and Peng [The Electronic J. Combin. 29 (4) (2022)] as well.

Perpetual cutoff method and discrete Ricci curvature bounds with exceptions

AbstractOne of the main obstacles regarding Bakry–Emery curvature on graphs is that the results require a global uniform lower curvature bounds where no exception sets are allowed. We overcome this obstacle by introducing the perpetual cutoff method. As applications, we prove gradient estimates only requiring curvature bounds on parts of the graph. Moreover, we sharply upper bound the distance to the exception set for graphs having uniformly positive Bakry–Emery curvature everywhere but on the exception set.

Region adjacency graph based GNN approach for static signature classification

Graph Neural Networks (GNNs) have gained popularity across various research fields in recent years. GNNs utilize graphs to construct an embedding that includes details about the nodes and edges in a graph’s neighborhood. In this work, a set of Region Adjacency Graphs (RAG) derives the attribute values from Static Signature (SS) images. These attribute values are used to label the nodes of the complete graph, which is formed by considering each signature as a node taken from the sample of signatures of a specific signer. The complete graph is trained by using GraphSAGE, an inductive representation learning method. This trained model helps to determine any newly introduced node (static signature to be tested) as genuine or fake. Standard static signature datasets, notably GPDSsynthetic and MCYT-75 are used to test the prevailing model. Experimental results on genuine and counterfeit signature networks demonstrate that our computed model enables a high rate of accuracy (GPDSsynthetic 99.91% and MCYT-75 99.56%) and minimum range of loss (GPDSsynthetic 0.0061 and MCYT-75 0.0070) on node classification.

On Proper 2-Dominating Sets in Graphs

This paper introduces the concept of proper 2-dominating sets in graphs, providing a comprehensive characterisation of graphs that possess such sets. We give the necessary and sufficient conditions for a graph to have a proper 2-dominating set. Graphs with proper dominating sets can have a symmetric structure. Moreover, we estimate the bounds of the proper 2-domination number in the graphs with respect to the 2-domination and 3-domination numbers. We show that the cardinality of γ2¯-set is greater by one at most than the cardinality of γ2-set.

A Graph Machine Learning Framework to Compute Zero Forcing Sets in Graphs

A Graph Machine Learning Framework to Compute Zero Forcing Sets in Graphs

Proposed crystal structure of carbadox, C11H10N4O4

A model for the crystal structure of carbadox has been generated and refined using synchrotron X-ray powder diffraction data and optimized using density functional theory techniques. Carbadox crystallizes in space group P21 (#4) with a = 13.8155(3), b = 21.4662(1), c = 16.3297(3) Å, β = 110.0931(7)°, V = 4548.10(3) Å3, and Z = 16. The crystal structure is characterized by approximately parallel stacking of the eight independent carbadox molecules parallel to the bc-plane. There are two different molecular configurations of the eight carbadox molecules; five are in the lower-energy configuration and three are in a ~10% higher-energy configuration. This arrangement likely achieves the lowest-energy crystalline packing via hydrogen bonding. Hydrogen bonds link the molecules both within and between the planes. Each of the amino groups forms a N–H⋯O hydrogen bond to an oxygen atom of the 1,4-dioxidoquinoxaline ring system of another molecule. The result is four pairs of hydrogen-bonded molecules, which form rings with graph set R2,2(14). Variation in specimen preparation can affect the preferred orientation of particles considerably. The powder pattern has been submitted to ICDD for inclusion in the Powder Diffraction File™ (PDF®).

Multi-part cross-intersecting families

Let A⊆[n]a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}\\subseteq {[n]\\atopwithdelims ()a}$$\\end{document} and B⊆[n]b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}\\subseteq {[n]\\atopwithdelims ()b}$$\\end{document} be two families of subsets of [n], we say A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}$$\\end{document} and B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}$$\\end{document} are cross-intersecting if A∩B≠∅\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\cap B\ e \\emptyset $$\\end{document} for all A∈A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\in {\\mathcal {A}}$$\\end{document}, B∈B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B\\in {\\mathcal {B}}$$\\end{document}. In this paper, we study cross-intersecting families in the multi-part setting. By characterizing the independent sets of vertex-transitive graphs and their direct products, we determine the sizes and structures of maximum-sized multi-part cross-intersecting families. This generalizes the results of Hilton’s (J Lond Math Soc 15(2):369–376, 1977) and Frankl–Tohushige’s (J Comb Theory Ser A 61(1):87–97, 1992) on cross-intersecting families in the single-part setting.

An iterative spectral algorithm for digraph clustering

Abstract Graph clustering is a fundamental technique in data analysis with applications in many different fields. While there is a large body of work on clustering undirected graphs, the problem of clustering directed graphs is much less understood. The analysis is more complex in the directed graph case for two reasons: the clustering must preserve directional information in the relationships between clusters, and directed graphs have non-Hermitian adjacency matrices whose properties are less conducive to traditional spectral methods. Here, we consider the problem of partitioning the vertex set of a directed graph into k≥2 clusters so that edges between different clusters tend to follow the same direction. We present an iterative algorithm based on spectral methods applied to new Hermitian representations of directed graphs. Our algorithm performs favourably against the state-of-the-art, both on synthetic and real-world data sets. Additionally, it can identify a ‘meta-graph’ of k vertices that represents the higher-order relations between clusters in a directed graph. We showcase this capability on data sets about food webs, biological neural networks, and the online card game Hearthstone.

Submodular Functions and Perfect Graphs

We give a combinatorial polynomial-time algorithm to find a maximum weight independent set in perfect graphs of bounded degree that do not contain a prism or a hole of length four as an induced subgraph. An even pair in a graph is a pair of vertices all induced paths between which are even. An even set is a set of vertices every two of which are an even pair. We show that every perfect graph that does not contain a prism or a hole of length four as an induced subgraph has a balanced separator which is the union of a bounded number of even sets, where the bound depends only on the maximum degree of the graph. This allows us to solve the maximum weight independent set problem using the well-known submodular function minimization algorithm. Funding: This work was supported by the Engineering and Physical Sciences Research Council [Grant EP/V002813/1]; the National Science Foundation [Grants DMS-1763817, DMS-2120644, and DMS-2303251]; and Alexander von Humboldt-Stiftung.