In this paper, a new class of rings, called FIC rings, is introduced for studying quasi-zero-divisor graphs of rings. Let R be a ring. The quasi-zero-divisor graph of R, denoted by Г＊（R）, is a directed graph defin...In this paper, a new class of rings, called FIC rings, is introduced for studying quasi-zero-divisor graphs of rings. Let R be a ring. The quasi-zero-divisor graph of R, denoted by Г＊（R）, is a directed graph defined on its nonzero quasi-zero-divisors, where there is an arc from a vertex x to another vertex y if and only if xRy = 0. We show that the following three conditions on an FIC ring R are equivalent： （1） χ（R） is finite; （2） ω（R） is finite; （3） Nil＊ R is finite where Nil.R equals the finite intersection of prime ideals. Furthermore, we also completely determine the connectedness, the diameter and the girth of Г＊ （R）.展开更多
In [1], Joe Warfel investigated the diameter of a zero-divisor graph for a direct product R 1 × R 2 with respect to the diameter of the zero-divisor graph of R 1 and R 2 . But the author only considered those gra...In [1], Joe Warfel investigated the diameter of a zero-divisor graph for a direct product R 1 × R 2 with respect to the diameter of the zero-divisor graph of R 1 and R 2 . But the author only considered those graphs whose diameters ≥ 1 and discussed six cases. This paper further discusses the other nine cases and also gives a complete characterization for the possible diameters for left Artin rings.展开更多
This paper introduces an ideal-boyed zero-divisor graph of non-commutative rings,denoted ΓI（R）.ΓI（R） is a directed graph.The properties and possible structures of the graph is studied.
We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that...We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that the maximal right quotient ring of a potent semiprimitive normal ring is abelian regular, moreover, the zero-divisor graph of such a ring is studied.展开更多
Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by , is a graph with vertices in , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a an...Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by , is a graph with vertices in , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in are adjacent if and only if and . In this paper, we investigate some combinatorial properties of the cozero-divisor graphs and such as connectivity, diameter, girth, clique numbers and planarity. We also study the cozero-divisor graphs of the direct products of two arbitrary commutative rings.展开更多
Chemical compounds are modeled as graphs.The atoms of molecules represent the graph vertices while chemical bonds between the atoms express the edges.The topological indices representing the molecular graph correspond...Chemical compounds are modeled as graphs.The atoms of molecules represent the graph vertices while chemical bonds between the atoms express the edges.The topological indices representing the molecular graph corresponds to the different chemical properties of compounds.Let a,b be are two positive integers,andΓ(Z_(a)×Z_(b))be the zero-divisor graph of the commutative ring Z_(a)×Z_(b).In this article some direct questions have been answered that can be utilized latterly in different applications.This study starts with simple computations,leading to a quite complex ring theoretic problems to prove certain properties.The theory of finite commutative rings is useful due to its different applications in the fields of advanced mechanics,communication theory,cryptography,combinatorics,algorithms analysis,and engineering.In this paper we determine the distance-based topological polynomials and indices of the zero-divisor graph of the commutative ring Z_(p^(2))×Z_(q)(for p,q as prime numbers)with the help of graphical structure analysis.The study outcomes help in understanding the fundamental relation between ring-theoretic and graph-theoretic properties of a zero-divisor graphΓ(G).展开更多
In this paper,we define signed zero-divisor graphs over commutative rings and investigate the interplay between the algebraic properties of the rings and the combinatorial properties of their corresponding signed zero...In this paper,we define signed zero-divisor graphs over commutative rings and investigate the interplay between the algebraic properties of the rings and the combinatorial properties of their corresponding signed zero-divisor graphs.We investigate the structure of signed zero-divisor graphs,the relation between ideals and signed zero-divisor graphs,and the adjacency matrices and the spectra of signed zero-divisor graphs.展开更多
In the international shipping industry, digital intelligence transformation has become essential, with both governments and enterprises actively working to integrate diverse datasets. The domain of maritime and shippi...In the international shipping industry, digital intelligence transformation has become essential, with both governments and enterprises actively working to integrate diverse datasets. The domain of maritime and shipping is characterized by a vast array of document types, filled with complex, large-scale, and often chaotic knowledge and relationships. Effectively managing these documents is crucial for developing a Large Language Model (LLM) in the maritime domain, enabling practitioners to access and leverage valuable information. A Knowledge Graph (KG) offers a state-of-the-art solution for enhancing knowledge retrieval, providing more accurate responses and enabling context-aware reasoning. This paper presents a framework for utilizing maritime and shipping documents to construct a knowledge graph using GraphRAG, a hybrid tool combining graph-based retrieval and generation capabilities. The extraction of entities and relationships from these documents and the KG construction process are detailed. Furthermore, the KG is integrated with an LLM to develop a Q&A system, demonstrating that the system significantly improves answer accuracy compared to traditional LLMs. Additionally, the KG construction process is up to 50% faster than conventional LLM-based approaches, underscoring the efficiency of our method. This study provides a promising approach to digital intelligence in shipping, advancing knowledge accessibility and decision-making.展开更多
The increasing popularity of the Internet and the widespread use of information technology have led to a rise in the number and sophistication of network attacks and security threats.Intrusion detection systems are cr...The increasing popularity of the Internet and the widespread use of information technology have led to a rise in the number and sophistication of network attacks and security threats.Intrusion detection systems are crucial to network security,playing a pivotal role in safeguarding networks from potential threats.However,in the context of an evolving landscape of sophisticated and elusive attacks,existing intrusion detection methodologies often overlook critical aspects such as changes in network topology over time and interactions between hosts.To address these issues,this paper proposes a real-time network intrusion detection method based on graph neural networks.The proposedmethod leverages the advantages of graph neural networks and employs a straightforward graph construction method to represent network traffic as dynamic graph-structured data.Additionally,a graph convolution operation with a multi-head attention mechanism is utilized to enhance the model’s ability to capture the intricate relationships within the graph structure comprehensively.Furthermore,it uses an integrated graph neural network to address dynamic graphs’structural and topological changes at different time points and the challenges of edge embedding in intrusion detection data.The edge classification problem is effectively transformed into node classification by employing a line graph data representation,which facilitates fine-grained intrusion detection tasks on dynamic graph node feature representations.The efficacy of the proposed method is evaluated using two commonly used intrusion detection datasets,UNSW-NB15 and NF-ToN-IoT-v2,and results are compared with previous studies in this field.The experimental results demonstrate that our proposed method achieves 99.3%and 99.96%accuracy on the two datasets,respectively,and outperforms the benchmark model in several evaluation metrics.展开更多
With the emphasis on user privacy and communication security, encrypted traffic has increased dramatically, which brings great challenges to traffic classification. The classification method of encrypted traffic based...With the emphasis on user privacy and communication security, encrypted traffic has increased dramatically, which brings great challenges to traffic classification. The classification method of encrypted traffic based on GNN can deal with encrypted traffic well. However, existing GNN-based approaches ignore the relationship between client or server packets. In this paper, we design a network traffic topology based on GCN, called Flow Mapping Graph (FMG). FMG establishes sequential edges between vertexes by the arrival order of packets and establishes jump-order edges between vertexes by connecting packets in different bursts with the same direction. It not only reflects the time characteristics of the packet but also strengthens the relationship between the client or server packets. According to FMG, a Traffic Mapping Classification model (TMC-GCN) is designed, which can automatically capture and learn the characteristics and structure information of the top vertex in FMG. The TMC-GCN model is used to classify the encrypted traffic. The encryption stream classification problem is transformed into a graph classification problem, which can effectively deal with data from different data sources and application scenarios. By comparing the performance of TMC-GCN with other classical models in four public datasets, including CICIOT2023, ISCXVPN2016, CICAAGM2017, and GraphDapp, the effectiveness of the FMG algorithm is verified. The experimental results show that the accuracy rate of the TMC-GCN model is 96.13%, the recall rate is 95.04%, and the F1 rate is 94.54%.展开更多
A graph G is H-free,if it contains no H as a subgraph.A graph G is said to be H-minor free,if it does not contain H as a minor.In 2010,Nikiforov asked that what the maximum spectral radius of an H-free graph of order ...A graph G is H-free,if it contains no H as a subgraph.A graph G is said to be H-minor free,if it does not contain H as a minor.In 2010,Nikiforov asked that what the maximum spectral radius of an H-free graph of order n is.In this paper,we consider some Brualdi-Solheid-Turan type problems on bipartite graphs.In 2015,Zhai,Lin and Gong in[Linear Algebra Appl.,2015,471:21-27]proved that if G is a bipartite graph with order n≥2k+2 and ρ(G)≥ρ(K_(k,n-k)),then G contains a C_(2k+2) unless G≌K_(k,n-k).First,we give a new and more simple proof for the above theorem.Second,we prove that if G is a bipartite graph with order n≥2k+2 and ρ(G)≥ρ(K_(k,n-k)),then G contains all T_(2k+3) unless G≌K_(k,n-k).Finally,we prove that among all outerplanar bipartite graphs on n≥308026 vertices,K_(1,n-1) attains the maximum spectral radius.展开更多
Spectrum-based fault localization (SBFL) generates a ranked list of suspicious elements by using the program execution spectrum, but the excessive number of elements ranked in parallel results in low localization accu...Spectrum-based fault localization (SBFL) generates a ranked list of suspicious elements by using the program execution spectrum, but the excessive number of elements ranked in parallel results in low localization accuracy. Most researchers consider intra-class dependencies to improve localization accuracy. However, some studies show that inter-class method call type faults account for more than 20%, which means such methods still have certain limitations. To solve the above problems, this paper proposes a two-phase software fault localization based on relational graph convolutional neural networks (Two-RGCNFL). Firstly, in Phase 1, the method call dependence graph (MCDG) of the program is constructed, the intra-class and inter-class dependencies in MCDG are extracted by using the relational graph convolutional neural network, and the classifier is used to identify the faulty methods. Then, the GraphSMOTE algorithm is improved to alleviate the impact of class imbalance on classification accuracy. Aiming at the problem of parallel ranking of element suspicious values in traditional SBFL technology, in Phase 2, Doc2Vec is used to learn static features, while spectrum information serves as dynamic features. A RankNet model based on siamese multi-layer perceptron is constructed to score and rank statements in the faulty method. This work conducts experiments on 5 real projects of Defects4J benchmark. Experimental results show that, compared with the traditional SBFL technique and two baseline methods, our approach improves the Top-1 accuracy by 262.86%, 29.59% and 53.01%, respectively, which verifies the effectiveness of Two-RGCNFL. Furthermore, this work verifies the importance of inter-class dependencies through ablation experiments.展开更多
The concept of matching energy was proposed by Gutman and Wagner firstly in 2012. Let G be a simple graph of order n and λ1, λ2, . . . , λn be the zeros of its matching polynomial. The matching energy of a graph G ...The concept of matching energy was proposed by Gutman and Wagner firstly in 2012. Let G be a simple graph of order n and λ1, λ2, . . . , λn be the zeros of its matching polynomial. The matching energy of a graph G is defined as ME(G) = Pni=1 |λi|. By the famous Coulson’s formula, matching energies can also be calculated by an improper integral depending on a parameter. A k-claw attaching graph Gu(k) refers to the graph obtained by attaching k pendent edges to the graph G at the vertex u, where u is called the root of Gu(k). In this paper, we use some theories of mathematical analysis to obtain a new technique to compare the matching energies of two k-claw attaching graphs Gu(k) and Hv(k) with the same order, that is, limk→∞[ME(Gu(k)) − ME(Hv(k))] = ME(G − u) − ME(H − v). By the technique, we finally determine unicyclic graphs of order n with the 9th to 13th minimal matching energies for all n ≥ 58.展开更多
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 u...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.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.1137134311161006+4 种基金1166101411171142)the Guangxi Science Research and Technology Development Project(Grant No.1599005-2-13)the Scientic Research Fund of Guangxi Education Department(Grant No.KY2015ZD075)the Natural Science Foundation of Guangxi(Grant No.2016GXSFDA380017)
文摘In this paper, a new class of rings, called FIC rings, is introduced for studying quasi-zero-divisor graphs of rings. Let R be a ring. The quasi-zero-divisor graph of R, denoted by Г＊（R）, is a directed graph defined on its nonzero quasi-zero-divisors, where there is an arc from a vertex x to another vertex y if and only if xRy = 0. We show that the following three conditions on an FIC ring R are equivalent： （1） χ（R） is finite; （2） ω（R） is finite; （3） Nil＊ R is finite where Nil.R equals the finite intersection of prime ideals. Furthermore, we also completely determine the connectedness, the diameter and the girth of Г＊ （R）.
基金Supported by the Natural Sciences Foundation of Guangxi Province(0575052, 0640070)Supported by the Innovation Project of Guangxi Graduate Education(2006106030701M05)Supported by the Scientific Research Foundation of Guangxi Educational Committee(200707LX233
文摘In [1], Joe Warfel investigated the diameter of a zero-divisor graph for a direct product R 1 × R 2 with respect to the diameter of the zero-divisor graph of R 1 and R 2 . But the author only considered those graphs whose diameters ≥ 1 and discussed six cases. This paper further discusses the other nine cases and also gives a complete characterization for the possible diameters for left Artin rings.
基金Supported by Guangxi Natural Sciences Foundation(0575052,0640070)Supported byInnovation Project of Guangxi Graduate Education(2006106030701M05)Supported Scientific Research Foun-dation of Guangxi Educational Committee
文摘This paper introduces an ideal-boyed zero-divisor graph of non-commutative rings,denoted ΓI（R）.ΓI（R） is a directed graph.The properties and possible structures of the graph is studied.
基金Partially supported by the NSF (10071035) of China.
文摘We introduce the zero-divisor graph for an abelian regular ring and show that if R,S are abelian regular, then (K0(R),[R])≌(K0(S),[S]) if and only if they have isomorphic reduced zero-divisor graphs. It is shown that the maximal right quotient ring of a potent semiprimitive normal ring is abelian regular, moreover, the zero-divisor graph of such a ring is studied.
文摘Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by , is a graph with vertices in , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in are adjacent if and only if and . In this paper, we investigate some combinatorial properties of the cozero-divisor graphs and such as connectivity, diameter, girth, clique numbers and planarity. We also study the cozero-divisor graphs of the direct products of two arbitrary commutative rings.
文摘Chemical compounds are modeled as graphs.The atoms of molecules represent the graph vertices while chemical bonds between the atoms express the edges.The topological indices representing the molecular graph corresponds to the different chemical properties of compounds.Let a,b be are two positive integers,andΓ(Z_(a)×Z_(b))be the zero-divisor graph of the commutative ring Z_(a)×Z_(b).In this article some direct questions have been answered that can be utilized latterly in different applications.This study starts with simple computations,leading to a quite complex ring theoretic problems to prove certain properties.The theory of finite commutative rings is useful due to its different applications in the fields of advanced mechanics,communication theory,cryptography,combinatorics,algorithms analysis,and engineering.In this paper we determine the distance-based topological polynomials and indices of the zero-divisor graph of the commutative ring Z_(p^(2))×Z_(q)(for p,q as prime numbers)with the help of graphical structure analysis.The study outcomes help in understanding the fundamental relation between ring-theoretic and graph-theoretic properties of a zero-divisor graphΓ(G).
基金supported by NSFC(Grant No.12001544)Hunan Provincial Natural Science Foundation(Grant No.2021JJ40707)+1 种基金supported by NSFC(Grant Nos.11871479,12071484)Hunan Provincial Natural Science Foundation(Grant Nos.2018JJ2479,2020JJ4675).
文摘In this paper,we define signed zero-divisor graphs over commutative rings and investigate the interplay between the algebraic properties of the rings and the combinatorial properties of their corresponding signed zero-divisor graphs.We investigate the structure of signed zero-divisor graphs,the relation between ideals and signed zero-divisor graphs,and the adjacency matrices and the spectra of signed zero-divisor graphs.
文摘In the international shipping industry, digital intelligence transformation has become essential, with both governments and enterprises actively working to integrate diverse datasets. The domain of maritime and shipping is characterized by a vast array of document types, filled with complex, large-scale, and often chaotic knowledge and relationships. Effectively managing these documents is crucial for developing a Large Language Model (LLM) in the maritime domain, enabling practitioners to access and leverage valuable information. A Knowledge Graph (KG) offers a state-of-the-art solution for enhancing knowledge retrieval, providing more accurate responses and enabling context-aware reasoning. This paper presents a framework for utilizing maritime and shipping documents to construct a knowledge graph using GraphRAG, a hybrid tool combining graph-based retrieval and generation capabilities. The extraction of entities and relationships from these documents and the KG construction process are detailed. Furthermore, the KG is integrated with an LLM to develop a Q&A system, demonstrating that the system significantly improves answer accuracy compared to traditional LLMs. Additionally, the KG construction process is up to 50% faster than conventional LLM-based approaches, underscoring the efficiency of our method. This study provides a promising approach to digital intelligence in shipping, advancing knowledge accessibility and decision-making.
文摘The increasing popularity of the Internet and the widespread use of information technology have led to a rise in the number and sophistication of network attacks and security threats.Intrusion detection systems are crucial to network security,playing a pivotal role in safeguarding networks from potential threats.However,in the context of an evolving landscape of sophisticated and elusive attacks,existing intrusion detection methodologies often overlook critical aspects such as changes in network topology over time and interactions between hosts.To address these issues,this paper proposes a real-time network intrusion detection method based on graph neural networks.The proposedmethod leverages the advantages of graph neural networks and employs a straightforward graph construction method to represent network traffic as dynamic graph-structured data.Additionally,a graph convolution operation with a multi-head attention mechanism is utilized to enhance the model’s ability to capture the intricate relationships within the graph structure comprehensively.Furthermore,it uses an integrated graph neural network to address dynamic graphs’structural and topological changes at different time points and the challenges of edge embedding in intrusion detection data.The edge classification problem is effectively transformed into node classification by employing a line graph data representation,which facilitates fine-grained intrusion detection tasks on dynamic graph node feature representations.The efficacy of the proposed method is evaluated using two commonly used intrusion detection datasets,UNSW-NB15 and NF-ToN-IoT-v2,and results are compared with previous studies in this field.The experimental results demonstrate that our proposed method achieves 99.3%and 99.96%accuracy on the two datasets,respectively,and outperforms the benchmark model in several evaluation metrics.
基金supported by the National Key Research and Development Program of China No.2023YFA1009500.
文摘With the emphasis on user privacy and communication security, encrypted traffic has increased dramatically, which brings great challenges to traffic classification. The classification method of encrypted traffic based on GNN can deal with encrypted traffic well. However, existing GNN-based approaches ignore the relationship between client or server packets. In this paper, we design a network traffic topology based on GCN, called Flow Mapping Graph (FMG). FMG establishes sequential edges between vertexes by the arrival order of packets and establishes jump-order edges between vertexes by connecting packets in different bursts with the same direction. It not only reflects the time characteristics of the packet but also strengthens the relationship between the client or server packets. According to FMG, a Traffic Mapping Classification model (TMC-GCN) is designed, which can automatically capture and learn the characteristics and structure information of the top vertex in FMG. The TMC-GCN model is used to classify the encrypted traffic. The encryption stream classification problem is transformed into a graph classification problem, which can effectively deal with data from different data sources and application scenarios. By comparing the performance of TMC-GCN with other classical models in four public datasets, including CICIOT2023, ISCXVPN2016, CICAAGM2017, and GraphDapp, the effectiveness of the FMG algorithm is verified. The experimental results show that the accuracy rate of the TMC-GCN model is 96.13%, the recall rate is 95.04%, and the F1 rate is 94.54%.
基金Supported by NSFC(No.12271162)Natural Science Foundation of Shanghai(No.22ZR1416300).
文摘A graph G is H-free,if it contains no H as a subgraph.A graph G is said to be H-minor free,if it does not contain H as a minor.In 2010,Nikiforov asked that what the maximum spectral radius of an H-free graph of order n is.In this paper,we consider some Brualdi-Solheid-Turan type problems on bipartite graphs.In 2015,Zhai,Lin and Gong in[Linear Algebra Appl.,2015,471:21-27]proved that if G is a bipartite graph with order n≥2k+2 and ρ(G)≥ρ(K_(k,n-k)),then G contains a C_(2k+2) unless G≌K_(k,n-k).First,we give a new and more simple proof for the above theorem.Second,we prove that if G is a bipartite graph with order n≥2k+2 and ρ(G)≥ρ(K_(k,n-k)),then G contains all T_(2k+3) unless G≌K_(k,n-k).Finally,we prove that among all outerplanar bipartite graphs on n≥308026 vertices,K_(1,n-1) attains the maximum spectral radius.
基金funded by the Youth Fund of the National Natural Science Foundation of China(Grant No.42261070).
文摘Spectrum-based fault localization (SBFL) generates a ranked list of suspicious elements by using the program execution spectrum, but the excessive number of elements ranked in parallel results in low localization accuracy. Most researchers consider intra-class dependencies to improve localization accuracy. However, some studies show that inter-class method call type faults account for more than 20%, which means such methods still have certain limitations. To solve the above problems, this paper proposes a two-phase software fault localization based on relational graph convolutional neural networks (Two-RGCNFL). Firstly, in Phase 1, the method call dependence graph (MCDG) of the program is constructed, the intra-class and inter-class dependencies in MCDG are extracted by using the relational graph convolutional neural network, and the classifier is used to identify the faulty methods. Then, the GraphSMOTE algorithm is improved to alleviate the impact of class imbalance on classification accuracy. Aiming at the problem of parallel ranking of element suspicious values in traditional SBFL technology, in Phase 2, Doc2Vec is used to learn static features, while spectrum information serves as dynamic features. A RankNet model based on siamese multi-layer perceptron is constructed to score and rank statements in the faulty method. This work conducts experiments on 5 real projects of Defects4J benchmark. Experimental results show that, compared with the traditional SBFL technique and two baseline methods, our approach improves the Top-1 accuracy by 262.86%, 29.59% and 53.01%, respectively, which verifies the effectiveness of Two-RGCNFL. Furthermore, this work verifies the importance of inter-class dependencies through ablation experiments.
基金Supported by the National Natural Science Foundation of China(Nos.12271439,11871398)the National College Students Innovation and Entrepreneurship Training Program(No.201910699173)。
文摘The concept of matching energy was proposed by Gutman and Wagner firstly in 2012. Let G be a simple graph of order n and λ1, λ2, . . . , λn be the zeros of its matching polynomial. The matching energy of a graph G is defined as ME(G) = Pni=1 |λi|. By the famous Coulson’s formula, matching energies can also be calculated by an improper integral depending on a parameter. A k-claw attaching graph Gu(k) refers to the graph obtained by attaching k pendent edges to the graph G at the vertex u, where u is called the root of Gu(k). In this paper, we use some theories of mathematical analysis to obtain a new technique to compare the matching energies of two k-claw attaching graphs Gu(k) and Hv(k) with the same order, that is, limk→∞[ME(Gu(k)) − ME(Hv(k))] = ME(G − u) − ME(H − v). By the technique, we finally determine unicyclic graphs of order n with the 9th to 13th minimal matching energies for all n ≥ 58.
文摘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.
