Theory of the Cayley graphs is directly linked with the group theory.However,if there are uncertainties on the vertices or edges or both then fuzzy graphs have an extraordinary importance.In this perspective,numbers o...Theory of the Cayley graphs is directly linked with the group theory.However,if there are uncertainties on the vertices or edges or both then fuzzy graphs have an extraordinary importance.In this perspective,numbers of generalηizations of fuzzy graphs have been explored in the literature.Among the others,picture fuzzy graph(PFG)has its own importance.A picture fuzzy graph(PFG)is a pair G=(C,D)defined on a H^(*)=(A,B),where C=(ηC,θ_(C),■_(C))is a picture fuzzy set on A and D=(ηD,θ_(D),■_(D))is a picture fuzzy set over the set B∈A×A such that for any edge mn∈ B with ηD(m,n)≤min(ηC(m),ηC(n)),θD(m,n)≤min(θC(m),θC(n))and ■_(D)(m,n)≥max(■_(C)(m),■_(C)(n)).In this manuscript,we introduce the notion of the Cayley picture fuzzy graphs on groups which is the generalization of the picture fuzzy graphs.Firstly,we discuss few important characteristics of the Cayley picture fuzzy graphs.We show that Cayley picture fuzzy graphs are vertex transitive and hence regular.Then,we investigate different types of Cayley graphs induced by the Cayley picture fuzzy graphs by using different types of cuts.We extensively discuss the term connectivity of the Cayley picture fuzzy graphs.Vertex connectivity and edge connectivity of the Cayley picture fuzzy graphs are also addressed.We also investigate the linkage between these two.Throughout,we provide the extensions of some characηteristics of both the PFGs and Cayley fuzzy graphs in the setting of Cayley picture fuzzy graphs.Finally,we provide the model of interconnected networks based on the Cayley picture fuzzy graphs.展开更多
Recently, the inverse connected p-median problem on block graphs G(V,E,w) under various cost functions, say rectilinear norm, Chebyshev norm, and bottleneck Hamming distance. Their contributions include finding a nece...Recently, the inverse connected p-median problem on block graphs G(V,E,w) under various cost functions, say rectilinear norm, Chebyshev norm, and bottleneck Hamming distance. Their contributions include finding a necessary and sufficient condition for the connected p-median problem on block graphs, developing algorithms and showing that these problems can be solved in O(n log n) time, where n is the number of vertices in the underlying block graph. Using similar technique, we show that some results are incorrect by a counter-example. Then we redefine some notations, reprove Theorem 1 and redescribe Theorem 2, Theorem 3 and Theorem 4.展开更多
Material exchange frequently occurs in gullies,and thus the relationship between a gullynetwork structure and sediment transport potential has attracted considerable interest.However,previous researches ignored the di...Material exchange frequently occurs in gullies,and thus the relationship between a gullynetwork structure and sediment transport potential has attracted considerable interest.However,previous researches ignored the difficulty of material transport from sources to sinks,and did not quantify the connectivity of a network structure.In this study,we used a graph model structure to model gully networks of six typical sample areas in the Loess Plateau of China and quantified gully network connectivity using four indexes:average node strength,accessibility from sources to sinks,potential flow,and network structural connectivity index.Results show that:(1)Reflected by different quantitative indexes,the trends of gully network connectivity in different regions are similar.From north to south,the connectivity of a sample area first increases and then decreases.(2)The more mature gullies have stronger network connectivity.Small resistance is conducive to material transport in the gullies.(3)The node connectivity index of the gully network shows a significant aggregation distribution in space,and node connectivity on the main channel is often stronger than that on the branch trench.These results not only deepen the understanding of the process and mechanism of loess gully geomorphic development and evolution but also provide a reference for geomorphic studies.展开更多
Let G be a k-connected graph, and T be a subset of V(G). If G-T is not connected,then T is said to be a cut-set of G. A k-cut-set T of G is a cut-set of G with │T│=k. Let T bea k-cut-set of a k-connected graph G. ...Let G be a k-connected graph, and T be a subset of V(G). If G-T is not connected,then T is said to be a cut-set of G. A k-cut-set T of G is a cut-set of G with │T│=k. Let T bea k-cut-set of a k-connected graph G. If G - T can be partitioned into subgraphs G1 and G2such that │G1│≥ 2, │G2│ 〉 2, then we call T a nontrivial k-cut-set of G. Suppose that G is a(k-1)-connected graph without nontrivial (k - 1)-cut-set. Then we call G a quasi k-connectedgraph. In this paper, we prove that for any integer k ≥ 5, if G is a k-connected graph withoutK4-, then every vertex of G is incident with an edge whose contraction yields a quasi k-connectedgraph, and so there are at least │V(G)│/2 edges of G such that the contraction of every member ofthem results in a quasi k-connected graph.展开更多
Patients with mild traumatic brain injury have a diverse clinical presentation,and the underlying pathophysiology remains poorly understood.Magnetic resonance imaging is a non-invasive technique that has been widely u...Patients with mild traumatic brain injury have a diverse clinical presentation,and the underlying pathophysiology remains poorly understood.Magnetic resonance imaging is a non-invasive technique that has been widely utilized to investigate neuro biological markers after mild traumatic brain injury.This approach has emerged as a promising tool for investigating the pathogenesis of mild traumatic brain injury.G raph theory is a quantitative method of analyzing complex networks that has been widely used to study changes in brain structure and function.However,most previous mild traumatic brain injury studies using graph theory have focused on specific populations,with limited exploration of simultaneous abnormalities in structural and functional connectivity.Given that mild traumatic brain injury is the most common type of traumatic brain injury encounte red in clinical practice,further investigation of the patient characteristics and evolution of structural and functional connectivity is critical.In the present study,we explored whether abnormal structural and functional connectivity in the acute phase could serve as indicators of longitudinal changes in imaging data and cognitive function in patients with mild traumatic brain injury.In this longitudinal study,we enrolled 46 patients with mild traumatic brain injury who were assessed within 2 wee ks of injury,as well as 36 healthy controls.Resting-state functional magnetic resonance imaging and diffusion-weighted imaging data were acquired for graph theoretical network analysis.In the acute phase,patients with mild traumatic brain injury demonstrated reduced structural connectivity in the dorsal attention network.More than 3 months of followup data revealed signs of recovery in structural and functional connectivity,as well as cognitive function,in 22 out of the 46 patients.Furthermore,better cognitive function was associated with more efficient networks.Finally,our data indicated that small-worldness in the acute stage could serve as a predictor of longitudinal changes in connectivity in patients with mild traumatic brain injury.These findings highlight the importance of integrating structural and functional connectivity in unde rstanding the occurrence and evolution of mild traumatic brain injury.Additionally,exploratory analysis based on subnetworks could serve a predictive function in the prognosis of patients with mild traumatic brain injury.展开更多
G. C. Ying, Y. Y. Meng, B. E. Sagan, and V. R. Vatter [1] found the maximum number of maximal independent sets in connected graphs which contain at most two cycles. In this paper, we give an alternative proof to deter...G. C. Ying, Y. Y. Meng, B. E. Sagan, and V. R. Vatter [1] found the maximum number of maximal independent sets in connected graphs which contain at most two cycles. In this paper, we give an alternative proof to determine the largest number of maximal independent sets among all connected graphs of order n ≥ 12, which contain at most two cycles. We also characterize the extremal graph achieving this maximum value.展开更多
Let G be a 3-connected graph with n vertices. The paper proves that if for each pair of vertices u and v of G, d(u,v)=2, has |N(u)∩N(v)|≤α(α is the minimum independent set number), and then max{d(u),d(v)}≥n+12,...Let G be a 3-connected graph with n vertices. The paper proves that if for each pair of vertices u and v of G, d(u,v)=2, has |N(u)∩N(v)|≤α(α is the minimum independent set number), and then max{d(u),d(v)}≥n+12, then G is a Hamilton connected graph.展开更多
如果 S 是一个统治集合,导致的 subgraph 至多有 k 部件, V 的子集 S 被称为一个连接 k 的统治集合。G 的连接 k 的支配数字 kc (G) 是在 G 的所有最小的连接 k 的统治集合上拿的最小的集的势。在这份报纸,我们与相等的连接支配和 2-...如果 S 是一个统治集合,导致的 subgraph 至多有 k 部件, V 的子集 S 被称为一个连接 k 的统治集合。G 的连接 k 的支配数字 kc (G) 是在 G 的所有最小的连接 k 的统治集合上拿的最小的集的势。在这份报纸,我们与相等的连接支配和 2-connected 支配数字描绘树和 unicyclic 图。展开更多
Tian and Meng in [Y. Tian and J. Meng, λc -Optimally half vertex transitive graphs with regularity k, Information Processing Letters 109 (2009) 683 - 686] shown that a connected half vertex transitive graph with regu...Tian and Meng in [Y. Tian and J. Meng, λc -Optimally half vertex transitive graphs with regularity k, Information Processing Letters 109 (2009) 683 - 686] shown that a connected half vertex transitive graph with regularity k and girth g(G) ≥ 6 is cyclically optimal. In this paper, we show that a connected half vertex transitive graph G is super cyclically edge-connected if minimum degree δ(G) ≥ 6 and girth g(G) ≥ 6.展开更多
It is proved that every 3 connected loopless multigraph has maximum genus at least one third of its cycle rank plus one if its cycle rank is not less than ten, and if its cycle rank is less than ten,it is upper emb...It is proved that every 3 connected loopless multigraph has maximum genus at least one third of its cycle rank plus one if its cycle rank is not less than ten, and if its cycle rank is less than ten,it is upper embeddable.This lower bound is tight.There are infinitely many 3 connected loopless multigraphs attaining this bound.展开更多
Let h be a nonnegative integer. The h-restricted edge connectivity λ h(G) of a simple connected graph G is defined as the minimum cardinality over the sets of edges of G, if any, whose removal disconnects G and every...Let h be a nonnegative integer. The h-restricted edge connectivity λ h(G) of a simple connected graph G is defined as the minimum cardinality over the sets of edges of G, if any, whose removal disconnects G and every component of the resulting graph has more than h vertices. This paper gave a necessary and sufficient condition and also three useful sufficient conditions to guarantee the existence of λ h(G). Moreover, it explicitly characterized the graphs whose 2-restricted edge connectivities do not exist.展开更多
The atom-bond connectivity(ABC) index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes,which is defined as ABC(G) =∑ uv∈E(G) √d u+dv-2 dudv,...The atom-bond connectivity(ABC) index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes,which is defined as ABC(G) =∑ uv∈E(G) √d u+dv-2 dudv,where du denotes the degree of a vertex u in G.A chemical graph is a graph in which no vertex has degree greater than 4.In this paper,we obtain the sharp upper and lower bounds on ABC index of chemical bicyclic graphs.展开更多
The eccentric connectivity index based on degree and eccentricity of the vertices of a graph is a widely used graph invariant in mathematics. In this paper we present the explicit generalized expressions for the eccen...The eccentric connectivity index based on degree and eccentricity of the vertices of a graph is a widely used graph invariant in mathematics. In this paper we present the explicit generalized expressions for the eccentric connectivity index and polynomial of the thorn graphs, and then consider some particular cases.展开更多
Let Gbe a connected k(≥3)-regulargraph w ith girth g. A setSofthe edgesin G is called an R2-edge-cutifG- Sis disconnected and contains neither an isolated vertex nor a one- degree vertex. The R2-edge-connectivity of ...Let Gbe a connected k(≥3)-regulargraph w ith girth g. A setSofthe edgesin G is called an R2-edge-cutifG- Sis disconnected and contains neither an isolated vertex nor a one- degree vertex. The R2-edge-connectivity of G, denoted by λ″(G), is the m inim um cardinality over allR2-edge-cuts, w hich is an im portantm easure for fault-tolerance of com puter intercon- nection netw orks. In this paper, λ″(G)= g(2k- 2) for any 2k-regular connected graph G(≠ K5) that is either edge-transitive or vertex-transitive and g≥5 is given.展开更多
A restricted edge cut is an edge cut of a connected graph whose removal resultsin a disconnected graph without isolated vertices. The size of a minimum restricted edge cutof a graph G is called its restricted edge con...A restricted edge cut is an edge cut of a connected graph whose removal resultsin a disconnected graph without isolated vertices. The size of a minimum restricted edge cutof a graph G is called its restricted edge connectivity, and is denoted by λ′(G). Let ξ(G) bethe minimum edge degree of graph G. It is known that λ′(G) ≤ξ(G) if G contains restrictededge cuts. Graph G is called maximal restricted edge connected if the equality holds in thethe preceding inequality. In this paper, undirected Kautz graph UK(2, n) is proved to bemaximal restricted edge connected if n ≥ 2.展开更多
The vertex connectivity k(G) of a graph G is the minimum number of nodes whose deletion disconnects it. Graph connectivity is one of the most fundamental problems in graph theory. In this paper, we designed an O(n2) t...The vertex connectivity k(G) of a graph G is the minimum number of nodes whose deletion disconnects it. Graph connectivity is one of the most fundamental problems in graph theory. In this paper, we designed an O(n2) time algorithm to solve connectivity problem on circular trapezoid graphs.展开更多
The topological connectivity information derived from the brain functional network can bring new insights for diagnosing and analyzing dementia disorders.The brain functional network is suitable to bridge the correlat...The topological connectivity information derived from the brain functional network can bring new insights for diagnosing and analyzing dementia disorders.The brain functional network is suitable to bridge the correlation between abnormal connectivities and dementia disorders.However,it is challenging to access considerable amounts of brain functional network data,which hinders the widespread application of data-driven models in dementia diagnosis.In this study,a novel distribution-regularized adversarial graph auto-Encoder(DAGAE)with transformer is proposed to generate new fake brain functional networks to augment the brain functional network dataset,improving the dementia diagnosis accuracy of data-driven models.Specifically,the label distribution is estimated to regularize the latent space learned by the graph encoder,which canmake the learning process stable and the learned representation robust.Also,the transformer generator is devised to map the node representations into node-to-node connections by exploring the long-term dependence of highly-correlated distant brain regions.The typical topological properties and discriminative features can be preserved entirely.Furthermore,the generated brain functional networks improve the prediction performance using different classifiers,which can be applied to analyze other cognitive diseases.Attempts on the Alzheimer’s Disease Neuroimaging Initiative(ADNI)dataset demonstrate that the proposed model can generate good brain functional networks.The classification results show adding generated data can achieve the best accuracy value of 85.33%,sensitivity value of 84.00%,specificity value of 86.67%.The proposed model also achieves superior performance compared with other related augmentedmodels.Overall,the proposedmodel effectively improves cognitive disease diagnosis by generating diverse brain functional networks.展开更多
文摘Theory of the Cayley graphs is directly linked with the group theory.However,if there are uncertainties on the vertices or edges or both then fuzzy graphs have an extraordinary importance.In this perspective,numbers of generalηizations of fuzzy graphs have been explored in the literature.Among the others,picture fuzzy graph(PFG)has its own importance.A picture fuzzy graph(PFG)is a pair G=(C,D)defined on a H^(*)=(A,B),where C=(ηC,θ_(C),■_(C))is a picture fuzzy set on A and D=(ηD,θ_(D),■_(D))is a picture fuzzy set over the set B∈A×A such that for any edge mn∈ B with ηD(m,n)≤min(ηC(m),ηC(n)),θD(m,n)≤min(θC(m),θC(n))and ■_(D)(m,n)≥max(■_(C)(m),■_(C)(n)).In this manuscript,we introduce the notion of the Cayley picture fuzzy graphs on groups which is the generalization of the picture fuzzy graphs.Firstly,we discuss few important characteristics of the Cayley picture fuzzy graphs.We show that Cayley picture fuzzy graphs are vertex transitive and hence regular.Then,we investigate different types of Cayley graphs induced by the Cayley picture fuzzy graphs by using different types of cuts.We extensively discuss the term connectivity of the Cayley picture fuzzy graphs.Vertex connectivity and edge connectivity of the Cayley picture fuzzy graphs are also addressed.We also investigate the linkage between these two.Throughout,we provide the extensions of some characηteristics of both the PFGs and Cayley fuzzy graphs in the setting of Cayley picture fuzzy graphs.Finally,we provide the model of interconnected networks based on the Cayley picture fuzzy graphs.
文摘Recently, the inverse connected p-median problem on block graphs G(V,E,w) under various cost functions, say rectilinear norm, Chebyshev norm, and bottleneck Hamming distance. Their contributions include finding a necessary and sufficient condition for the connected p-median problem on block graphs, developing algorithms and showing that these problems can be solved in O(n log n) time, where n is the number of vertices in the underlying block graph. Using similar technique, we show that some results are incorrect by a counter-example. Then we redefine some notations, reprove Theorem 1 and redescribe Theorem 2, Theorem 3 and Theorem 4.
基金supported by the National Natural Science Foundation of China(Grant Nos.42271421 and 41930102)。
文摘Material exchange frequently occurs in gullies,and thus the relationship between a gullynetwork structure and sediment transport potential has attracted considerable interest.However,previous researches ignored the difficulty of material transport from sources to sinks,and did not quantify the connectivity of a network structure.In this study,we used a graph model structure to model gully networks of six typical sample areas in the Loess Plateau of China and quantified gully network connectivity using four indexes:average node strength,accessibility from sources to sinks,potential flow,and network structural connectivity index.Results show that:(1)Reflected by different quantitative indexes,the trends of gully network connectivity in different regions are similar.From north to south,the connectivity of a sample area first increases and then decreases.(2)The more mature gullies have stronger network connectivity.Small resistance is conducive to material transport in the gullies.(3)The node connectivity index of the gully network shows a significant aggregation distribution in space,and node connectivity on the main channel is often stronger than that on the branch trench.These results not only deepen the understanding of the process and mechanism of loess gully geomorphic development and evolution but also provide a reference for geomorphic studies.
基金supported by National Natural Science Foundation of China(11071016)Union Foundation of The Science and Technology Department of Guizhou Province,Anshun GovernmentAnshun University(Qiankehe LH Zi[2014]7500)
文摘Let G be a k-connected graph, and T be a subset of V(G). If G-T is not connected,then T is said to be a cut-set of G. A k-cut-set T of G is a cut-set of G with │T│=k. Let T bea k-cut-set of a k-connected graph G. If G - T can be partitioned into subgraphs G1 and G2such that │G1│≥ 2, │G2│ 〉 2, then we call T a nontrivial k-cut-set of G. Suppose that G is a(k-1)-connected graph without nontrivial (k - 1)-cut-set. Then we call G a quasi k-connectedgraph. In this paper, we prove that for any integer k ≥ 5, if G is a k-connected graph withoutK4-, then every vertex of G is incident with an edge whose contraction yields a quasi k-connectedgraph, and so there are at least │V(G)│/2 edges of G such that the contraction of every member ofthem results in a quasi k-connected graph.
基金supported by the National Natural Science Foundation of China,Nos.81671671(to JL),61971451(to JL),U22A2034(to XK),62177047(to XK)the National Defense Science and Technology Collaborative Innovation Major Project of Central South University,No.2021gfcx05(to JL)+6 种基金Clinical Research Cen terfor Medical Imaging of Hunan Province,No.2020SK4001(to JL)Key Emergency Project of Pneumonia Epidemic of Novel Coronavirus Infection of Hu nan Province,No.2020SK3006(to JL)Innovative Special Construction Foundation of Hunan Province,No.2019SK2131(to JL)the Science and Technology lnnovation Program of Hunan Province,Nos.2021RC4016(to JL),2021SK53503(to ML)Scientific Research Program of Hunan Commission of Health,No.202209044797(to JL)Central South University Research Program of Advanced Interdisciplinary Studies,No.2023Q YJC020(to XK)the Natural Science Foundation of Hunan Province,No.2022JJ30814(to ML)。
文摘Patients with mild traumatic brain injury have a diverse clinical presentation,and the underlying pathophysiology remains poorly understood.Magnetic resonance imaging is a non-invasive technique that has been widely utilized to investigate neuro biological markers after mild traumatic brain injury.This approach has emerged as a promising tool for investigating the pathogenesis of mild traumatic brain injury.G raph theory is a quantitative method of analyzing complex networks that has been widely used to study changes in brain structure and function.However,most previous mild traumatic brain injury studies using graph theory have focused on specific populations,with limited exploration of simultaneous abnormalities in structural and functional connectivity.Given that mild traumatic brain injury is the most common type of traumatic brain injury encounte red in clinical practice,further investigation of the patient characteristics and evolution of structural and functional connectivity is critical.In the present study,we explored whether abnormal structural and functional connectivity in the acute phase could serve as indicators of longitudinal changes in imaging data and cognitive function in patients with mild traumatic brain injury.In this longitudinal study,we enrolled 46 patients with mild traumatic brain injury who were assessed within 2 wee ks of injury,as well as 36 healthy controls.Resting-state functional magnetic resonance imaging and diffusion-weighted imaging data were acquired for graph theoretical network analysis.In the acute phase,patients with mild traumatic brain injury demonstrated reduced structural connectivity in the dorsal attention network.More than 3 months of followup data revealed signs of recovery in structural and functional connectivity,as well as cognitive function,in 22 out of the 46 patients.Furthermore,better cognitive function was associated with more efficient networks.Finally,our data indicated that small-worldness in the acute stage could serve as a predictor of longitudinal changes in connectivity in patients with mild traumatic brain injury.These findings highlight the importance of integrating structural and functional connectivity in unde rstanding the occurrence and evolution of mild traumatic brain injury.Additionally,exploratory analysis based on subnetworks could serve a predictive function in the prognosis of patients with mild traumatic brain injury.
文摘G. C. Ying, Y. Y. Meng, B. E. Sagan, and V. R. Vatter [1] found the maximum number of maximal independent sets in connected graphs which contain at most two cycles. In this paper, we give an alternative proof to determine the largest number of maximal independent sets among all connected graphs of order n ≥ 12, which contain at most two cycles. We also characterize the extremal graph achieving this maximum value.
文摘Let G be a 3-connected graph with n vertices. The paper proves that if for each pair of vertices u and v of G, d(u,v)=2, has |N(u)∩N(v)|≤α(α is the minimum independent set number), and then max{d(u),d(v)}≥n+12, then G is a Hamilton connected graph.
文摘如果 S 是一个统治集合,导致的 subgraph 至多有 k 部件, V 的子集 S 被称为一个连接 k 的统治集合。G 的连接 k 的支配数字 kc (G) 是在 G 的所有最小的连接 k 的统治集合上拿的最小的集的势。在这份报纸,我们与相等的连接支配和 2-connected 支配数字描绘树和 unicyclic 图。
文摘Tian and Meng in [Y. Tian and J. Meng, λc -Optimally half vertex transitive graphs with regularity k, Information Processing Letters 109 (2009) 683 - 686] shown that a connected half vertex transitive graph with regularity k and girth g(G) ≥ 6 is cyclically optimal. In this paper, we show that a connected half vertex transitive graph G is super cyclically edge-connected if minimum degree δ(G) ≥ 6 and girth g(G) ≥ 6.
文摘It is proved that every 3 connected loopless multigraph has maximum genus at least one third of its cycle rank plus one if its cycle rank is not less than ten, and if its cycle rank is less than ten,it is upper embeddable.This lower bound is tight.There are infinitely many 3 connected loopless multigraphs attaining this bound.
基金National Natural Science Foundation of China( No.199710 5 6)
文摘Let h be a nonnegative integer. The h-restricted edge connectivity λ h(G) of a simple connected graph G is defined as the minimum cardinality over the sets of edges of G, if any, whose removal disconnects G and every component of the resulting graph has more than h vertices. This paper gave a necessary and sufficient condition and also three useful sufficient conditions to guarantee the existence of λ h(G). Moreover, it explicitly characterized the graphs whose 2-restricted edge connectivities do not exist.
基金Supported by the National Natural Science Foundation of China(11071272,10831001,11171279,11101087)the Young Talent Foundation of Fuzhou University(XRC-1154)
文摘The atom-bond connectivity(ABC) index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes,which is defined as ABC(G) =∑ uv∈E(G) √d u+dv-2 dudv,where du denotes the degree of a vertex u in G.A chemical graph is a graph in which no vertex has degree greater than 4.In this paper,we obtain the sharp upper and lower bounds on ABC index of chemical bicyclic graphs.
文摘The eccentric connectivity index based on degree and eccentricity of the vertices of a graph is a widely used graph invariant in mathematics. In this paper we present the explicit generalized expressions for the eccentric connectivity index and polynomial of the thorn graphs, and then consider some particular cases.
文摘Let Gbe a connected k(≥3)-regulargraph w ith girth g. A setSofthe edgesin G is called an R2-edge-cutifG- Sis disconnected and contains neither an isolated vertex nor a one- degree vertex. The R2-edge-connectivity of G, denoted by λ″(G), is the m inim um cardinality over allR2-edge-cuts, w hich is an im portantm easure for fault-tolerance of com puter intercon- nection netw orks. In this paper, λ″(G)= g(2k- 2) for any 2k-regular connected graph G(≠ K5) that is either edge-transitive or vertex-transitive and g≥5 is given.
基金Supported by the NNSF of China(10271105) Supported by the NSF of Fujian EducationMinistry(JA03145) Supported by the NNSF of China(10071080)
文摘A restricted edge cut is an edge cut of a connected graph whose removal resultsin a disconnected graph without isolated vertices. The size of a minimum restricted edge cutof a graph G is called its restricted edge connectivity, and is denoted by λ′(G). Let ξ(G) bethe minimum edge degree of graph G. It is known that λ′(G) ≤ξ(G) if G contains restrictededge cuts. Graph G is called maximal restricted edge connected if the equality holds in thethe preceding inequality. In this paper, undirected Kautz graph UK(2, n) is proved to bemaximal restricted edge connected if n ≥ 2.
文摘The vertex connectivity k(G) of a graph G is the minimum number of nodes whose deletion disconnects it. Graph connectivity is one of the most fundamental problems in graph theory. In this paper, we designed an O(n2) time algorithm to solve connectivity problem on circular trapezoid graphs.
基金This paper is partially supported by the British Heart Foundation Accelerator Award,UK(AA\18\3\34220)Royal Society International Exchanges Cost Share Award,UK(RP202G0230)+9 种基金Hope Foundation for Cancer Research,UK(RM60G0680)Medical Research Council Confidence in Concept Award,UK(MC_PC_17171)Sino-UK Industrial Fund,UK(RP202G0289)Global Challenges Research Fund(GCRF),UK(P202PF11)LIAS Pioneering Partnerships Award,UK(P202ED10)Data Science Enhancement Fund,UK(P202RE237)Fight for Sight,UK(24NN201)Sino-UK Education Fund,UK(OP202006)Biotechnology and Biological Sciences Research Council,UK(RM32G0178B8)LIAS Seed Corn,UK(P202RE969).
文摘The topological connectivity information derived from the brain functional network can bring new insights for diagnosing and analyzing dementia disorders.The brain functional network is suitable to bridge the correlation between abnormal connectivities and dementia disorders.However,it is challenging to access considerable amounts of brain functional network data,which hinders the widespread application of data-driven models in dementia diagnosis.In this study,a novel distribution-regularized adversarial graph auto-Encoder(DAGAE)with transformer is proposed to generate new fake brain functional networks to augment the brain functional network dataset,improving the dementia diagnosis accuracy of data-driven models.Specifically,the label distribution is estimated to regularize the latent space learned by the graph encoder,which canmake the learning process stable and the learned representation robust.Also,the transformer generator is devised to map the node representations into node-to-node connections by exploring the long-term dependence of highly-correlated distant brain regions.The typical topological properties and discriminative features can be preserved entirely.Furthermore,the generated brain functional networks improve the prediction performance using different classifiers,which can be applied to analyze other cognitive diseases.Attempts on the Alzheimer’s Disease Neuroimaging Initiative(ADNI)dataset demonstrate that the proposed model can generate good brain functional networks.The classification results show adding generated data can achieve the best accuracy value of 85.33%,sensitivity value of 84.00%,specificity value of 86.67%.The proposed model also achieves superior performance compared with other related augmentedmodels.Overall,the proposedmodel effectively improves cognitive disease diagnosis by generating diverse brain functional networks.