Hypergraph Research Articles

Optimal embedding of hypercube into exchanged hypercube and optical multi-mesh hypercube

Embedding a (guest) graph into (host) another graph can be utilized for structural simulation, processor allocation, and algorithm porting in the field of parallel architecture. Dilation and edge congestion are two prominent indicators which evaluate the quality of the embedding. In this paper, we consider the guest graph as hypercube network and the host graphs to be exchanged hypercube and optical multi-mesh hypercube. We propose new ordering which minimizes dilation and edge congestion. We also compute the layout of embedding hypercube into the aforementioned interconnection networks.

MGHCN: Multi-graph structures and hypergraph convolutional networks for traffic flow prediction

Accurate and timely traffic flow predictions are essential for effective traffic management and congestion reduction. However, most traditional prediction methods often fail to capture the complex dynamics and correlations within traffic flows due to insufficient processing of spatiotemporal data. Specifically, these methods struggle to integrate and analyze the multi-layered spatial and temporal interactions inherent in traffic data, leading to suboptimal prediction accuracy and robustness. To address this limitation, this paper presents a Multi-Graph Structures and Hypergraph Convolutional Network (MGHCN) that combines diverse graphs and hypergraphs. The MGHCN simplifies the predictive framework by integrating key components that improve its robustness and accuracy. One of the most critical components is the dual hypergraph structure, which captures edge correlations by converting traditional graph edges into hypergraph nodes. To better capture the spatiotemporal correlation of traffic data, a Graph Convolutional Network (GCN) is employed to analyze these hypergraphs in depth. Finally, a novel adjacency matrix and a dynamic graph module are used to accurately simulate interactions between spatiotemporal features, thereby enhancing the accuracy and robustness of predictions. Experimental validation on four distinct real-world traffic datasets shows that MGHCN outperforms existing state-of-the-art traffic prediction methods.

SPLHRNMTF: robust orthogonal non-negative matrix tri-factorization with self-paced learning and dual hypergraph regularization for predicting miRNA-disease associations

MicroRNAs (miRNAs) have been demonstrated to be closely related to human diseases. Studying the potential associations between miRNAs and diseases contributes to our understanding of disease pathogenic mechanisms. As traditional biological experiments are costly and time-consuming, computational models can be considered as effective complementary tools. In this study, we propose a novel model of robust orthogonal non-negative matrix tri-factorization (NMTF) with self-paced learning and dual hypergraph regularization, named SPLHRNMTF, to predict miRNA-disease associations. More specifically, SPLHRNMTF first uses a non-linear fusion method to obtain miRNA and disease comprehensive similarity. Subsequently, the improved miRNA-disease association matrix is reformulated based on weighted k-nearest neighbor profiles to correct false-negative associations. In addition, we utilize L2,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_{2,1}$$\\end{document} norm to replace Frobenius norm to calculate residual error, alleviating the impact of noise and outliers on prediction performance. Then, we integrate self-paced learning into NMTF to alleviate the model from falling into bad local optimal solutions by gradually including samples from easy to complex. Finally, hypergraph regularization is introduced to capture high-order complex relations from hypergraphs related to miRNAs and diseases. In 5-fold cross-validation five times experiments, SPLHRNMTF obtains higher average AUC values than other baseline models. Moreover, the case studies on breast neoplasms and lung neoplasms further demonstrate the accuracy of SPLHRNMTF. Meanwhile, the potential associations discovered are of biological significance.

DHCL-BR: Dual Hypergraph Contrastive Learning for Bundle Recommendation

Abstract As an extension of conventional top-K item recommendation solution, bundle recommendation has aroused increasingly attention. However, because of the extreme sparsity of user-bundle (UB) interactions, the existing top-K item recommendation methods suffer from poor performance when applied to bundle recommendation. While some graph-based approaches have been proposed for bundle recommendation, these approaches primarily leverage the bipartite graph to model the UB interactions, resulting in suboptimal performance. In this paper, a dual hypergraph contrastive learning model is proposed for bundle recommendation. First, we model the direct and indirect UB interactions as hypergraphs to represent the higher-order UB relations. Second, we utilize the hypergraph convolution networks to learn the user and bundle embeddings from the hypergraphs, and improve the learned embeddings through a bidirectional contrastive learning strategy. Finally, we adopt a joint loss that combines the InfoBPR loss supporting multiple negative samples and the contrastive losses to optimize model parameters for prediction. Experiments on the real-world datasets indicate that our model performs better than the state-of-the-art baseline methods.

Dual hypergraphs with feature weighted and latent space learning for the diagnosis of Alzheimer’s disease

In recent years of research on the diagnosis of Alzheimer’s disease, capturing data relationships can help improve model performance. However, the simple graph structure can only capture pairwise relationships between data, which cannot model the complex data relationships in real situations. In addition, the redundant features and noise from the original data space also harm the model performance. Dual hypergraphs with feature-weighted and latent space learning (DHFWLSL) method for Alzheimer’s disease diagnosis are proposed to solve the abovementioned problems. Hypergraphs are used to capture higher-order data relationships and construct hypergraphs for both sample and feature relationships to fully utilize the auxiliary ability of data relationships for model classification. In order to remove noise and redundant features from the original data space, latent space learning is utilized to project the original data into the latent space, and then data is projected into the label space for diagnosis. Finally, a feature-weighted strategy helps to improve the model performance further. An alternating optimization algorithm with the proof of convergence is used to solve the proposed model. Sufficient experiments support the validity and advantages of the proposed model.

Light dual hypergraph convolution for collaborative filtering

Recommender systems filter information to meet users’ personalized interests actively. Existing graph-based models typically extract users’ interests from a heterogeneous interaction graph. They do not distinguish learning between users and items, ignoring the heterogeneous property. In addition, the interaction sparsity and long-tail bias issues still limit the recommendation performance significantly. Fortunately, hidden homogeneous correlations that have a considerable volume can entangle abundant CF signals. In this paper, we propose a light dual hypergraph convolution (LDHC) for collaborative filtering, which designs a hypergraph to involve heterogeneous and homogeneous correlations with more CF signals confronting the challenges. Over the integrated hypergraph, a two-level interest propagation is performed within the heterogeneous interaction graph and between the homogeneous user/item graphs to model users’ interests, where learning on users and items is distinguished and collaborated by the homogeneous propagation. Specifically, hypergraph convolution is lightened by removing unnecessary parameters to propagate users’ interests. Extensive experiments on publicly available datasets demonstrate that the proposed LDHC outperforms the state-of-the-art baselines.

Effective Bounds for Induced Size-Ramsey Numbers of Cycles

The induced size-Ramsey number r^indk(H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{r}_\ ext {ind}^k(H)$$\\end{document} of a graph H is the smallest number of edges a (host) graph G can have such that for any k-coloring of its edges, there exists a monochromatic copy of H which is an induced subgraph of G. In 1995, in their seminal paper, Haxell, Kohayakawa and Łuczak showed that for cycles, these numbers are linear for any constant number of colours, i.e., r^indk(Cn)≤Cn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{r}_\ ext {ind}^k(C_n)\\le Cn$$\\end{document} for some C=C(k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C=C(k)$$\\end{document}. The constant C comes from the use of the regularity lemma, and has a tower type dependence on k. In this paper we significantly improve these bounds, showing that r^indk(Cn)≤O(k102)n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{r}_\ ext {ind}^k(C_n)\\le O(k^{102})n$$\\end{document} when n is even, thus obtaining only a polynomial dependence of C on k. We also prove r^indk(Cn)≤eO(klogk)n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{r}_\ ext {ind}^k(C_n)\\le e^{O(k\\log k)}n$$\\end{document} for odd n, which almost matches the lower bound of eΩ(k)n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^{\\Omega (k)}n$$\\end{document}. Finally, we show that the ordinary (non-induced) size-Ramsey number satisfies r^k(Cn)=eO(k)n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{r}^k(C_n)=e^{O(k)}n$$\\end{document} for odd n. This substantially improves the best previous result of eO(k2)n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^{O(k^2)}n$$\\end{document}, and is best possible, up to the implied constant in the exponent. To achieve our results, we present a new host graph construction which, roughly speaking, reduces our task to finding a cycle of approximate given length in a graph with local sparsity.

Minimum Congestion and Wirelength of Embedding the nth Cartesian Product of K4 into Various Kinds of Grid Networks

Embedding an interconnection network into another network is one of the main problems in parallel processing and computing systems. Interconnection networks in multiprocessing systems facilitate effective communication among various system components. Obtaining the minimum cutwidth, congestion, and wirelength in graph embedding problems is of great significance for network design and architecture, where the minimum is taken over all embedding of guest graphs into host graphs, and addressing these issues can reduce time and cost in embedded design. This work aims to accurately calculate the minimum congestion and wirelength for the [Formula: see text] (the [Formula: see text]th Cartesian product of [Formula: see text]) layout into [Formula: see text]-dimensional, [Formula: see text]-dimensional, and [Formula: see text]-dimensional grids by the optimal solution of edge isoperimetric problem on [Formula: see text].

The Weisfeiler-Leman Dimension of Conjunctive Queries

A graph parameter is a function f on graphs with the property that, for any pair of isomorphic graphs G 1 and G 2 , f(G 1 )=f(G 2 ). The Weisfeiler--Leman (WL) dimension of f is the minimum k such that, if G 1 and G 2 are indistinguishable by the k-dimensional WL-algorithm then f(G 1 )=f(G 2 ). The WL-dimension of f is ∞ if no such k exists. We study the WL-dimension of graph parameters characterised by the number of answers from a fixed conjunctive query to the graph. Given a conjunctive query φ, we quantify the WL-dimension of the function that maps every graph G to the number of answers of φ in G. The works of Dvorak (J. Graph Theory 2010), Dell, Grohe, and Rattan (ICALP 2018), and Neuen (ArXiv 2023) have answered this question for full conjunctive queries, which are conjunctive queries without existentially quantified variables. For such queries φ, the WL-dimension is equal to the treewidth of the Gaifman graph of φ. In this work, we give a characterisation that applies to all conjunctive queries. Given any conjunctive query φ, we prove that its WL-dimension is equal to the semantic extension width sew(φ), a novel width measure that can be thought of as a combination of the treewidth of φ and its quantified star size, an invariant introduced by Durand and Mengel (ICDT 2013) describing how the existentially quantified variables of φ are connected with the free variables. Using the recently established equivalence between the WL-algorithm and higher-order Graph Neural Networks (GNNs) due to Morris et al. (AAAI 2019), we obtain as a consequence that the function counting answers to a conjunctive query φ cannot be computed by GNNs of order smaller than sew(φ). The majority of the paper is concerned with establishing a lower bound of the WL-dimension of a query. Given any conjunctive query φ with semantic extension width k, we consider a graph F of treewidth k obtained from the Gaifman graph of φ by repeatedly cloning the vertices corresponding to existentially quantified variables. Using a modification due to Furer (ICALP 2001) of the Cai-Fürer-Immerman construction (Combinatorica 1992), we then obtain a pair of graphs χ(F) and ^χ(F) that are indistinguishable by the (k-1)-dimensional WL-algorithm since F has treewidth k. Finally, in the technical heart of the paper, we show that φ has a different number of answers in χ(F) and ^χ(F). Thus, φ can distinguish two graphs that cannot be distinguished by the (k-1)-dimensional WL-algorithm, so the WL-dimension of φ is at least k.

DHyper: A Recurrent Dual Hypergraph Neural Network for Event Prediction in Temporal Knowledge Graphs

Event prediction is a vital and challenging task in temporal knowledge graphs (TKGs), which have played crucial roles in various applications. Recently, many graph neural networks based approaches are proposed to model the graph structure information in TKGs. However, these approaches only construct graphs based on quadruplets and model the pairwise correlation between entities, which fail to capture the high-order correlations among entities. To this end, we propose DHyper, a recurrent Dual Hypergraph neural network for event prediction in TKGs, which simultaneously models the influences of the high-order correlations among both entities and relations. Specifically, a dual hypergraph learning module is proposed to discover the high-order correlations among entities and among relations in a parameterized way. A dual hypergraph message passing network is introduced to perform the information aggregation and representation fusion on the entity hypergraph and the relation hypergraph. Extensive experiments on six real-world datasets demonstrate that DHyper achieves the state-of-the-art performances, outperforming the best baseline by an average of 13.09%, 4.26%, 17.60%, and 18.03% in MRR, Hits@1, Hits@3, and Hits@10, respectively.

Framelet-based dual hypergraph neural networks for student performance prediction

Framelet-based dual hypergraph neural networks for student performance prediction

Hub Parameters of Hypergraph

A hypergraph is an extension of a graph in which one edge can connect any number of vertices. In contrary, an edge connects exactly two vertices in a graph.In this paper we introduce hub-hyperpath, hubset and hubnumber of a hypergraph. Also we defined the hubnumber of a different types of hypergraphs and analyze some of its properties. Then the hub number of a hypergraph is compared with its dual hypergraph. Hubset can be useful for various tasks, such as targeted marketing and social networks. Additionally, finding the hub number of a graph is useful in network security, as it helps in identifying nodes that, if compromised, could have a significant impact on the overall network.

Isoperimetric Inequalities and Supercritical Percolation on High-Dimensional Graphs

It is known that many different types of finite random subgraph models undergo quantitatively similar phase transitions around their percolation thresholds, and the proofs of these results rely on isoperimetric properties of the underlying host graph. Recently, the authors showed that such a phase transition occurs in a large class of regular high-dimensional product graphs, generalising a classic result for the hypercube. In this paper we give new isoperimetric inequalities for such regular high-dimensional product graphs, which generalise the well-known isoperimetric inequality of Harper for the hypercube, and are asymptotically sharp for a wide range of set sizes. We then use these isoperimetric properties to investigate the structure of the giant component L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} in supercritical percolation on these product graphs, that is, when p=1+ϵd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=\\frac{1+\\epsilon }{d}$$\\end{document}, where d is the degree of the product graph and ϵ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon >0$$\\end{document} is a small enough constant. We show that typically L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} has edge-expansion Ω1dlnd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\left( \\frac{1}{d\\ln d}\\right) $$\\end{document}. Furthermore, we show that L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} likely contains a linear-sized subgraph with vertex-expansion Ω1dlnd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\left( \\frac{1}{d\\ln d}\\right) $$\\end{document}. These results are best possible up to the logarithmic factor in d. Using these likely expansion properties, we determine, up to small polylogarithmic factors in d, the likely diameter of L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} as well as the typical mixing time of a lazy random walk on L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document}. Furthermore, we show the likely existence of a cycle of length Ωndlnd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\left( \\frac{n}{d\\ln d}\\right) $$\\end{document}. These results not only generalise, but also improve substantially upon the known bounds in the case of the hypercube, where in particular the likely diameter and typical mixing time of L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} were previously only known to be polynomial in d.

The Irrelevance of Influencers: Information Diffusion with Re-Activation and Immunity Lasts Exponentially Long on Social Network Models

Information diffusion models on networks are at the forefront of AI research. The dynamics of such models typically follow stochastic models from epidemiology, used to model not only infections but various phenomena, including the behavior of computer viruses and viral marketing campaigns. A core question in this setting is how to efficiently detect the most influential vertices in the host graph such that the infection survives the longest. In processes that incorporate re-infection of the vertices, such as the SIS process, theoretical studies identify parameter thresholds where the survival time of the process rapidly transitions from logarithmic to super-polynomial. These results contradict the intuition that the starting configuration is relevant, since the process will always either die out fast or survive almost indefinitely. A shortcoming of these results is that models incorporating short-term immunity (or creative advertisement fatigue) have not been subjected to such a theoretical analysis so far. We reduce this gap in the literature by studying the SIRS process, a more realistic model, which besides re-infection additionally incorporates short-term immunity. On complex network models, we identify parameter regimes for which the process survives exponentially long, and we get a tight threshold for random graphs. Underlying these results is our main technical contribution, showing a threshold behavior for the survival time of the SIRS process on graphs with large expander subgraphs, such as social network models.

Optimization of layout for embedding half hypercube into conventional tree architectures

SummaryEmbedding a graph into another graph can be utilized for structural simulation, processor allocation, and algorithm porting in the field of parallel architecture. This has the potential to enhance the physical layout of network‐on‐chip (NoC) devices as well as to investigate their virtualization possibilities. Layout is one of the many indicators of graph embedding. An optimal layout in NoC design can result in a decreased wiring area and cost, as well as the reduction in communication delay between parallel processing components. In this work, the guest graph is the half hypercube, which possess efficient routing, fault tolerance, and Hamiltonian features. The edge isoperimetric problems is solved for the half hypercube using a novel technique called the isochronal ordering. The host graphs considered in our work are complete binary trees, ‐rooted complete binary trees, and other few predominantly investigated tree based architectures.

Percolation on irregular high-dimensional product graphs

AbstractWe consider bond percolation on high-dimensional product graphs $G=\square _{i=1}^tG^{(i)}$ , where $\square$ denotes the Cartesian product. We call the $G^{(i)}$ the base graphs and the product graph $G$ the host graph. Very recently, Lichev (J. Graph Theory, 99(4):651–670, 2022) showed that, under a mild requirement on the isoperimetric properties of the base graphs, the component structure of the percolated graph $G_p$ undergoes a phase transition when $p$ is around $\frac{1}{d}$ , where $d$ is the average degree of the host graph.In the supercritical regime, we strengthen Lichev’s result by showing that the giant component is in fact unique, with all other components of order $o(|G|)$ , and determining the sharp asymptotic order of the giant. Furthermore, we answer two questions posed by Lichev (J. Graph Theory, 99(4):651–670, 2022): firstly, we provide a construction showing that the requirement of bounded degree is necessary for the likely emergence of a linear order component; secondly, we show that the isoperimetric requirement on the base graphs can be, in fact, super-exponentially small in the dimension. Finally, in the subcritical regime, we give an example showing that in the case of irregular high-dimensional product graphs, there can be a polynomially large component with high probability, very much unlike the quantitative behaviour seen in the Erdős-Rényi random graph and in the percolated hypercube, and in fact in any regular high-dimensional product graphs, as shown by the authors in a companion paper (Percolation on high-dimensional product graphs. arXiv:2209.03722, 2022).

Ramsey-Turán type results for matchings in edge colored graphs

Let G1,G2,…,Gc be arbitrary graphs. By G→(G1,G2,…,Gc), we mean if the edges of G are partitioned into c disjoint color classes giving c graphs H1,H2,…,Hc, then at least one Hi has a subgraph isomorphic to Gi. The Ramsey number R(G1,G2,…,Gc) is the smallest positive integer n such that Kn→(G1,G2,…,Gc). In this paper, we extend the result of Cockayne and Lorimer on the Ramsey number of stripes allowing host graphs with the corresponding Ore-type condition: If t≥1 and n1,n2,…,nc, n1=max⁡{n1,n2,…,nc}, are positive integers and G is a graph with at least max⁡{t,n1}+∑i=1c(ni−1)+1 vertices such that for each pair of non-adjacent vertices, the sum of the number of their non-neighbors is at most 2max⁡{t,n1}−1, then G→(n1K2,n2K2,…,ncK2). Consequently, we prove that if G is a graph with chromatic number at least max⁡{t,n1}+∑i=1c(ni−1)+1, then G→(St,n1K2,n2K2,…,ncK2), where St is the star graph with t edges. In addition, for given graph G, the Ramsey closure of G, denoted by C(G), is defined and we prove that if G is a graph with at least max⁡{t,n1}+∑i=1c(ni−1)+1 vertices, then G→(n1K2,n2K2,…,ncK2) if and only if C(G)→(n1K2,n2K2,…,ncK2). Thus, a condition that forces C(G)→(n1K2,n2K2,…,ncK2) also forces G→(n1K2,n2K2,…,ncK2). As a result, a condition is provided on the degree sequence of a graph G such that under this condition G→(n1K2,n2K2,…,ncK2).Further, this article provides a sharp bound for the maximum number of edges in a simple graph G such that G↛(n1K2,n2K2,…,ncK2). We also establish the uniqueness of such extremal graphs. This result provides a far-reaching generalization of an important classical result of Erdős and Gallai and also giving a new proof for the graph case of the conjecture of Erdős dating back to 1965, known as the Erdős' matching conjecture.

Alzheimer's disease classification based on nonlinear high-order features and hypergraph convolutional neural network

Alzheimer's disease (AD) is an irreversible neurodegenerative disorder that damages patients' memory and cognitive abilities. Therefore, the diagnosis of AD holds significant importance. The interactions between regions of interest (ROIs) in the brain often involve multiple areas collaborating in a nonlinear manner. Leveraging these nonlinear higher-order interaction features to their fullest potential contributes to enhancing the accuracy of AD diagnosis. To address this, a framework combining nonlinear higher-order feature extraction and three-dimensional (3D) hypergraph neural networks is proposed for computer-assisted diagnosis of AD. First, a support vector machine regression model based on the radial basis function kernel was trained on ROI data to obtain a base estimator. Then, a recursive feature elimination algorithm based on the base estimator was applied to extract nonlinear higher-order features from functional magnetic resonance imaging (fMRI) data. These features were subsequently constructed into a hypergraph, leveraging the complex interactions captured in the data. Finally, a four-dimensional (4D) spatiotemporal hypergraph convolutional neural network model was constructed based on the fMRI data for classification. Experimental results on the Alzheimer's Disease Neuroimaging Initiative (ADNI) database demonstrated that the proposed framework outperformed the Hyper Graph Convolutional Network (HyperGCN) framework by 8% and traditional two-dimensional (2D) linear feature extraction methods by 12% in the AD/normal control (NC) classification task. In conclusion, this framework demonstrates an improvement in AD classification compared to mainstream deep learning methods, providing valuable evidence for computer-assisted diagnosis of AD.

New Ideas in Recognition of Cancer and Neutrosophic Super Hyper Graph by Eulerian-Path-Cut as Hyper Eulogy-Path-Cut on Super EULA-Path-Cut

New Ideas in Recognition of Cancer and Neutrosophic Super Hyper Graph by Eulerian-Path-Cut as Hyper Eulogy-Path-Cut on Super EULA-Path-Cut

Partitioning a 2-edge-coloured graph of minimum degree [formula omitted] into three monochromatic cycles

Lehel conjectured in the 1970s that the vertices of every red and blue edge-coloured complete graph can be partitioned into two monochromatic cycles. This was confirmed in 2010 by Bessy and Thomassé. However, the host graph G does not have to be complete. It suffices to require that G has minimum degree at least 3n/4, where n is the order of G, as was shown recently by Letzter, confirming a conjecture of Balogh, Barát, Gerbner, Gyárfás and Sárközy. This degree condition is tight.Here we continue this line of research, by proving that for every red and blue edge-colouring of an n-vertex graph of minimum degree at least 2n/3+o(n), there is a partition of the vertex set into three monochromatic cycles. This approximately verifies a conjecture of Pokrovskiy and is essentially tight.