Hamiltonicity,neighborhood union and square graphs of claw-free graphs

Let G be a graph, the square graph G 2 of G is a graph satisfying V(G 2)=V(G) and E(G 2)=E(G)∪｛uv: dist G(u, v)=2｝ . In this paper, we use the technique of vertex insertion on l -connected ( l=k or k+1, k≥2 ) claw-free graphs to provide a unified proof for G to be Hamiltonian, 1 -Hamiltonian or Hamiltonian-connected. The sufficient conditions are expressed by the inequality concerning ∑ k i=0N(Y i) and n(Y) in G for each independent set Y=｛y 0, y 1, …, y k｝ of the square graph of G , where b ( 0<b<k+1 ) is an integer, Y i=｛y i, y i-1, …, y i-(b-1)｝Y for i∈｛0, 1, …, k｝ , where subscriptions of y j s will be taken modulo k+1 , and n(Y)=｛v∈ V(G): dist (v, Y)≤ 2｝ .

Neighborhood Union of Essential Sets and Hamiltonicity of Claw-Free Graphs

Let G be a graph, an independent set Y in G is called an essential independent set (or essential set for simplicity), if there is {y 1,y 2} Y such that dist (y 1,y 2)=2. In this paper, we will use the technique of the vertex insertion on l connected ( l=k or k+1,k≥2 ) claw free graphs to provide a unified proof for G to be hamiltonian or 1 hamiltonian, the sufficient conditions are expressed by the inequality concerning ∑ki=0N(Y i) and n(Y) for each essential set Y={y 0,y 1,...,y k} of G , where Y i={y i,y i-1 ,...,y i-(b-1) }Y for i∈{0,1,...,k} (the subscriptions of y j ’s will be taken modulo k+1 ), b ( 0【b【k+1 ) is an integer, and n(Y)={v∈V(G): dist (v,Y)≤2 }.

On hamiltonicity of 2-connected claw-free graphs

A graph G has the hourglass property if every induced hourglass S（a tree with a degree sequence 22224） contains two non-adjacent vertices which have a common neighbor in G-V（S）.For an integer k≥4,a graph G has the single k-cycle property if every edge of G,which does not lie in a triangle,lies in a cycle C of order at most k such that C has at least「｜V（C） /2」 edges which do not lie in a triangle,and they are not adjacent.In this paper,we show that every hourglass-free claw-free graph G of δ（G） ≥3 with the single 7-cycle property is Hamiltonian and is best possible;we also show that every claw-free graph G of δ（G） ≥3 with the hourglass property and with single 6-cycle property is Hamiltonian.

NEIGHBORHOOD UNION OF INDEPENDENT SETS AND HAMILTONICITY OF CLAW-FREE GRAPHS

Let G be a graph,for any u∈V(G),let N(u) denote the neighborhood of u and d(u)=|N(u)| be the degree of u.For any UV(G),let N(U)=∪_~u∈U N(u), and d(U)=|N(U)|.A graph G is called claw-free if it has no induced subgraph isomorphic to K_~1,3 .One of the fundamental results concerning cycles in claw-free graphs is due to Tian Feng,et al.: Let G be a 2-connected claw-free graph of order n,and d(u)+d(v)+d(w)≥n-2 for every independent vertex set {u,v,w} of G, then G is Hamiltonian. It is proved that,for any three positive integers s,t and w,such that if G is a (s+t+w-1)-connected claw-free graph of order n,and d(S)+d(T)+d(W)>n-(s+t+w) for every three disjoint independent vertex sets S,T,W with |S|=s,|T|=t,|W|=w,and S∪T∪W is also independent,then G is Hamiltonian.Other related results are obtained too.

Hamiltonian Cycles in Regular 2-Connected Claw-Free Graphs

A known result by Jackson Bill is that every 2-connected k-regular graph on at most 3k vertices is Hamiltonian. In this paper,it is proved that every 2-connected k-regular claw-free graph on at most 5k(k≥10)vertices is Hamiltonian. Moreover, the bound 5k is best possible. A counterexample of a 2-connected k-regular claw-free non-Hamiltonian graph on 5k+1 vertices is given, and it is conjectured that every 3-connected k-regular claw-free graph on at most 12k-7 vertices is Hamiltonian.

A Property of Claw-free Graphs

In this paper we consider a property of claw-free graphs.We show that if d(u)+ d(v)≥ν(G)+2k+3,for every two nonadjacent vertices u and v,then G is 2k-vertex-deletable IM-extendable,whereν(G)=|V(G)|.And the bound is tight.

Longest Paths and Cycles in Connected Claw-Free Graphs

A graph is called claw-free if it does not contain a claw as its induced subgraph.In this paper, we prove the following results:1)If G is a 2-connected claw-free graph on n vertices,then for any vertex v and any two distinct vertices x and y in V(G)-{v},G has a path containing v and all neighbors of v and connecting x and y;2) Let C be the longest cycle in a 3-connected claw-free graph G and H a component of G-C,and if H is connected but not 2-connected,then there exist nonadjacent vertices u and v in H such that |V(C)|≥(3(d(u)+)d(v))-2.

Characterizations of Cycle-Forced 2-Connected Claw-Free Cubic Graphs

Let G be a graph and C be an arbitrary even cycle of G.The graph G is called a cycle-forced graph if G-V(C)has a unique perfect matching.When C is an arbitrary induced even cycle of G,G is called an induced-cycle-forced graph.If G-V(C)has no perfect matching,G is said to be cycle-bad.This paper gives characterizations of these three type of graphs in the class of 2-connected claw-free cubic graphs.

Neighborhood Intersections and Hamiltonian property in Claw-Free Graphs

We prove the following result: Let G be a 2 connected claw free graph of order n(n≥3) and connectivity k . If for any independent set S k+1 with cardinality k+1 , there exist u,v∈S k+1 , such that |N(u)∩N(v)|≥(n-2k)/4 ,then G is Hamiltonian.

Dynamic Multi-Graph Spatio-Temporal Graph Traffic Flow Prediction in Bangkok:An Application of a Continuous Convolutional Neural Network

The ability to accurately predict urban traffic flows is crucial for optimising city operations.Consequently,various methods for forecasting urban traffic have been developed,focusing on analysing historical data to understand complex mobility patterns.Deep learning techniques,such as graph neural networks(GNNs),are popular for their ability to capture spatio-temporal dependencies.However,these models often become overly complex due to the large number of hyper-parameters involved.In this study,we introduce Dynamic Multi-Graph Spatial-Temporal Graph Neural Ordinary Differential Equation Networks(DMST-GNODE),a framework based on ordinary differential equations(ODEs)that autonomously discovers effective spatial-temporal graph neural network(STGNN)architectures for traffic prediction tasks.The comparative analysis of DMST-GNODE and baseline models indicates that DMST-GNODE model demonstrates superior performance across multiple datasets,consistently achieving the lowest Root Mean Square Error(RMSE)and Mean Absolute Error(MAE)values,alongside the highest accuracy.On the BKK(Bangkok)dataset,it outperformed other models with an RMSE of 3.3165 and an accuracy of 0.9367 for a 20-min interval,maintaining this trend across 40 and 60 min.Similarly,on the PeMS08 dataset,DMST-GNODE achieved the best performance with an RMSE of 19.4863 and an accuracy of 0.9377 at 20 min,demonstrating its effectiveness over longer periods.The Los_Loop dataset results further emphasise this model’s advantage,with an RMSE of 3.3422 and an accuracy of 0.7643 at 20 min,consistently maintaining superiority across all time intervals.These numerical highlights indicate that DMST-GNODE not only outperforms baseline models but also achieves higher accuracy and lower errors across different time intervals and datasets.

Cover a 3-regular Claw-free Graph by Induced Matchings

The induced matching cover number of a graph G without isolated vertices, denoted by imc(G),is the minimum integer k such that G has k induced matchings {M1,M2,···,Mk}such that,V(M1)∪V(M2)∪···∪V(Mk)covers V(G).This paper shows that,if G is a 3-regular claw-free graph,then imc(G)∈{2,3}.

Graph Transformers研究进展综述

随着图结构数据在各种实际场景中的广泛应用,对其进行有效建模和处理的需求日益增加。Graph Transformers(GTs)作为一类使用Transformers处理图数据的模型,能够有效缓解传统图神经网络(GNN)中存在的过平滑和过挤压等问题,因此可以学习到更好的特征表示。根据对近年来GTs相关文献的研究,将现有的模型架构分为两类:第一类通过绝对编码和相对编码向Transformers中加入图的位置和结构信息,以增强Transformers对图结构数据的理解和处理能力;第二类根据不同的方式(串行、交替、并行)将GNN与Transformers进行结合,以充分利用两者的优势。介绍了GTs在信息安全、药物发现和知识图谱等领域的应用,对比总结了不同用途的模型及其优缺点。最后,从可扩展性、复杂图、更好的结合方式等方面分析了GTs未来研究面临的挑战。

基于Graph Transformer的半监督异配图表示学习模型

现有的图卷积网络(GCN)模型基于同配性假设,无法直接应用于异配图的表示学习,且许多异配图表示学习的研究工作受消息传递机制的限制,导致节点特征混淆和特征过度挤压而出现过平滑问题。针对这些问题,提出一种基于Graph Transformer的半监督异配图表示学习模型HPGT(HeteroPhilic Graph Transformer)。首先,使用度连接概率矩阵采样节点的路径邻域,再通过自注意力机制自适应地聚合路径上的节点异配连接模式,编码得到节点的结构信息,用节点的原始属性信息和结构信息构建Transformer层的自注意力模块;其次,将每个节点自身的隐层表示与它的邻域节点的隐层表示分离更新以避免节点通过自注意力模块聚合过量的自身信息,再把每个节点表示与它的邻域表示连接,得到单个Transformer层的输出,另外,将所有的Transformer层的输出跳连到最终的节点隐层表示以防止中间层信息丢失;最后,使用线性层和Softmax层将节点的隐层表示映射到节点的预测标签。实验结果表明,与无结构编码(SE)的模型相比,基于度连接概率的SE能为Transformer层的自注意力模块提供有效的偏差信息,HPGT平均准确率提升0.99%~11.98%;与对比模型相比,在异配数据集(Texas、Cornell、Wisconsin和Actor)上,模型节点分类准确率提升0.21%~1.69%,在同配数据集(Cora、CiteSeer和PubMed)上,节点分类准确率分别达到了0.8379、0.7467和0.8862。以上结果验证了HPGT具有较强的异配图表示学习能力,尤其适用于强异配图节点分类任务。

Graph neural network-based scheduling for multi-UAV-enabled communications in D2D networks

In this paper,we jointly design the power control and position dispatch for Multi-Unmanned Aerial Vehicle(UAV)-enabled communication in Device-to-Device(D2D)networks.Our objective is to maximize the total transmission rate of Downlink Users(DUs).Meanwhile,the Quality of Service(QoS)of all D2D users must be satisfied.We comprehensively considered the interference among D2D communications and downlink transmissions.The original problem is strongly non-convex,which requires high computational complexity for traditional optimization methods.And to make matters worse,the results are not necessarily globally optimal.In this paper,we propose a novel Graph Neural Networks(GNN)based approach that can map the considered system into a specific graph structure and achieve the optimal solution in a low complexity manner.Particularly,we first construct a GNN-based model for the proposed network,in which the transmission links and interference links are formulated as vertexes and edges,respectively.Then,by taking the channel state information and the coordinates of ground users as the inputs,as well as the location of UAVs and the transmission power of all transmitters as outputs,we obtain the mapping from inputs to outputs through training the parameters of GNN.Simulation results verified that the way to maximize the total transmission rate of DUs can be extracted effectively via the training on samples.Moreover,it also shows that the performance of proposed GNN-based method is better than that of traditional means.

GraphMLP-Mixer:基于图-多层感知机架构的高效多行为序列推荐方法

在多行为序列推荐领域,图神经网络(GNNs)虽被广泛应用,但存在局限性,如对序列间协同信号建模不足和处理长距离依赖性等问题.针对这些问题,提出了一种新的解决框架GraphMLP-Mixer.该框架首先构造全局物品图来增强模型对序列间协同信号的建模,然后将感知机-混合器架构与图神经网络结合,得到图-感知机混合器模型对用户兴趣进行充分挖掘.GraphMLP-Mixer具有2个显著优势:一是能够有效捕捉用户行为的全局依赖性,同时减轻信息过压缩问题;二是其时间与空间效率显著提高,其复杂度与用户交互行为的数量成线性关系,优于现有基于GNN多行为序列推荐模型.在3个真实的公开数据集上进行实验,大量的实验结果验证了GraphMLP-Mixer在处理多行为序列推荐问题时的有效性和高效性.

基于GraphSAGE网络的藏文短文本分类研究

文本分类是自然语言处理领域的重要研究方向,由于藏文数据的稀缺性、语言学特征抽取的复杂性、篇章结构的多样性等因素导致藏文文本分类任务进展缓慢。因此,该文以图神经作为基础模型进行改进。首先,在“音节-音节”“音节-文档”建模的基础上,融合文档特征,采用二元分类模型动态网络构建“文档-文档”边,以充分挖掘短文本的全局特征,增加滑动窗口,减少模型的计算复杂度并寻找最优窗口取值。其次,针对藏文短文本的音节稀疏性,首次引入GraphSAGE作为基础模型,并探究不同聚合方式在藏文短文本分类上的性能差异。最后,为捕获节点间关系的异质性,对邻居节点进行特征加权再平均池化以增强模型的特征提取能力。在TNCC标题文本数据集上,该文模型的分类准确率达到了62.50%,与传统GCN、原始GraphSAGE和预训练语言模型CINO相比,该方法在分类准确率上分别提高了2.56%、1%和2.4%。

A STABILITY RESULT FOR TRANSLATINGSPACELIKE GRAPHS IN LORENTZ MANIFOLDS

In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.

A Value for Games Defined on Graphs

Given a graph g=( V,A ) , we define a space of subgraphs M with the binary operation of union and the unique decomposition property into blocks. This space allows us to discuss a notion of minimal subgraphs (minimal coalitions) that are of interest for the game. Additionally, a partition of the game is defined in terms of the gain of each block, and subsequently, a solution to the game is defined based on distributing to each player (node and edge) present in each block a payment proportional to their contribution to the coalition.

Heterophilic Graph Neural Network Based on Spatial and Frequency Domain Adaptive Embedding Mechanism

Graph Neural Networks(GNNs)play a significant role in tasks related to homophilic graphs.Traditional GNNs,based on the assumption of homophily,employ low-pass filters for neighboring nodes to achieve information aggregation and embedding.However,in heterophilic graphs,nodes from different categories often establish connections,while nodes of the same category are located further apart in the graph topology.This characteristic poses challenges to traditional GNNs,leading to issues of“distant node modeling deficiency”and“failure of the homophily assumption”.In response,this paper introduces the Spatial-Frequency domain Adaptive Heterophilic Graph Neural Networks(SFA-HGNN),which integrates adaptive embedding mechanisms for both spatial and frequency domains to address the aforementioned issues.Specifically,for the first problem,we propose the“Distant Spatial Embedding Module”,aiming to select and aggregate distant nodes through high-order randomwalk transition probabilities to enhance modeling capabilities.For the second issue,we design the“Proximal Frequency Domain Embedding Module”,constructing adaptive filters to separate high and low-frequency signals of nodes,and introduce frequency-domain guided attention mechanisms to fuse the relevant information,thereby reducing the noise introduced by the failure of the homophily assumption.We deploy the SFA-HGNN on six publicly available heterophilic networks,achieving state-of-the-art results in four of them.Furthermore,we elaborate on the hyperparameter selection mechanism and validate the performance of each module through experimentation,demonstrating a positive correlation between“node structural similarity”,“node attribute vector similarity”,and“node homophily”in heterophilic networks.

Weighted Forwarding in Graph Convolution Networks for Recommendation Information Systems

Recommendation Information Systems(RIS)are pivotal in helping users in swiftly locating desired content from the vast amount of information available on the Internet.Graph Convolution Network(GCN)algorithms have been employed to implement the RIS efficiently.However,the GCN algorithm faces limitations in terms of performance enhancement owing to the due to the embedding value-vanishing problem that occurs during the learning process.To address this issue,we propose a Weighted Forwarding method using the GCN(WF-GCN)algorithm.The proposed method involves multiplying the embedding results with different weights for each hop layer during graph learning.By applying the WF-GCN algorithm,which adjusts weights for each hop layer before forwarding to the next,nodes with many neighbors achieve higher embedding values.This approach facilitates the learning of more hop layers within the GCN framework.The efficacy of the WF-GCN was demonstrated through its application to various datasets.In the MovieLens dataset,the implementation of WF-GCN in LightGCN resulted in significant performance improvements,with recall and NDCG increasing by up to+163.64%and+132.04%,respectively.Similarly,in the Last.FM dataset,LightGCN using WF-GCN enhanced with WF-GCN showed substantial improvements,with the recall and NDCG metrics rising by up to+174.40%and+169.95%,respectively.Furthermore,the application of WF-GCN to Self-supervised Graph Learning(SGL)and Simple Graph Contrastive Learning(SimGCL)also demonstrated notable enhancements in both recall and NDCG across these datasets.