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Linear arboricity of Cartesian products of graphs
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作者 陶昉昀 林文松 《Journal of Southeast University(English Edition)》 EI CAS 2013年第2期222-225,共4页
A linear forest is a forest whose components are paths. The linear arboricity la (G) of a graph G is the minimum number of linear forests which partition the edge set E(G) of G. The Cartesian product G□H of two g... A linear forest is a forest whose components are paths. The linear arboricity la (G) of a graph G is the minimum number of linear forests which partition the edge set E(G) of G. The Cartesian product G□H of two graphs G and H is defined as the graph with vertex set V(G□H) = {(u, v)| u ∈V(G), v∈V(H) } and edge set E(G□H) = { ( u, x) ( v, Y)|u=v and xy∈E(H), or uv∈E(G) and x=y}. Let Pm and Cm,, respectively, denote the path and cycle on m vertices and K, denote the complete graph on n vertices. It is proved that (Km□Pm)=[n+1/2]for m≥2,la(Km□Cm)=[n+2/2],and la(Km□Km)=[n+m-1/2]. The methods to decompose these graphs into linear forests are given in the proofs. Furthermore, the linear arboricity conjecture is true for these classes of graphs. 展开更多
关键词 linear forest linear arboricity cartesian product
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The Crossing Numbers of Cartesian Products of Stars with a 5-Vertex Graph
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作者 苏振华 黄元秋 《Journal of Mathematical Research and Exposition》 CSCD 2009年第4期580-586,共7页
In this paper, we compute the crossing number of a specific graph Hn, and then by contraction, we obtain the conclusion that cr(G13 × Sn) = 4[n/2] [n-1/2]+[n/2] . The result fills up the blank of the crossing ... In this paper, we compute the crossing number of a specific graph Hn, and then by contraction, we obtain the conclusion that cr(G13 × Sn) = 4[n/2] [n-1/2]+[n/2] . The result fills up the blank of the crossing numbers of Cartesian products of stars with all 5-vertex graphs presented by Marian Klesc. 展开更多
关键词 GRAPH DRAWING crossing number cartesian products star.
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The Crossing Number of the Cartesian Products of Wm with Pn 被引量:6
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作者 WANG Jing LV Sheng Xiang HUANG Yuan Qiu 《Journal of Mathematical Research and Exposition》 CSCD 2009年第2期362-366,共5页
Most results on crossing numbers of graphs focus on some special graphs, such as the Cartesian products of small graphs with path, star and cycle. In this paper, we obtain the crossing number formula of Cartesian prod... Most results on crossing numbers of graphs focus on some special graphs, such as the Cartesian products of small graphs with path, star and cycle. In this paper, we obtain the crossing number formula of Cartesian products of wheel Wm with path Pn for arbitrary m ≥ 3 and n ≥ 1. 展开更多
关键词 DRAWING crossing number WHEEL PATH cartesian product.
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Full Friendly Index Sets of Cartesian Products of Two Cycles 被引量:3
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作者 Wai chee SHIU 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2010年第7期1233-1244,共12页
Let G =(V, E) be a connected simple graph. A labeling f : V → Z2 induces an edge labeling f* : E → Z2 defined by f*(xy) = f(x) +f(y) for each xy ∈ E. For i ∈ Z2, let vf(i) = |f^-1(i)| and ef(i... Let G =(V, E) be a connected simple graph. A labeling f : V → Z2 induces an edge labeling f* : E → Z2 defined by f*(xy) = f(x) +f(y) for each xy ∈ E. For i ∈ Z2, let vf(i) = |f^-1(i)| and ef(i) = |f*^-1(i)|. A labeling f is called friendly if |vf(1) - vf(0)| ≤ 1. For a friendly labeling f of a graph G, we define the friendly index of G under f by if(G) = e(1) - el(0). The set [if(G) | f is a friendly labeling of G} is called the full friendly index set of G, denoted by FFI(G). In this paper, we will determine the full friendly index set of every Cartesian product of two cycles. 展开更多
关键词 vertex labeling friendly labeling friendly index set cartesian product of two cycles
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L(2, 1)-Circular Labelings of Cartesian Products of Complete Graphs 被引量:2
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作者 LV Da Mei LIN Wen Song SONG Zeng Min 《Journal of Mathematical Research and Exposition》 CSCD 2009年第1期91-98,共8页
For positive integers j and k with j ≥ k, an L(j, k)-labeling of a graph G is an assignment of nonnegative integers to V(G) such that the difference between labels of adjacent vertices is at least j, and the diff... For positive integers j and k with j ≥ k, an L(j, k)-labeling of a graph G is an assignment of nonnegative integers to V(G) such that the difference between labels of adjacent vertices is at least j, and the difference between labels of vertices that are distance two apart is at least k. The span of an L(j, k)-labeling of a graph G is the difference between the maximum and minimum integers it uses. The λj, k-number of G is the minimum span taken over all L(j, k)-labelings of G. An m-(j, k)-circular labeling of a graph G is a function f : V(G) →{0, 1, 2,..., m - 1} such that |f(u) - f(v)|m ≥ j if u and v are adjacent; and |f(u) - f(v)|m 〉 k ifu and v are at distance two, where |x|m = min{|xl|, m-|x|}. The minimum integer m such that there exists an m-(j, k)-circular labeling of G is called the σj,k-number of G and is denoted by σj,k(G). This paper determines the σ2,1-number of the Cartesian product of any three complete graphs. 展开更多
关键词 λ2 1-number σ2 1-number cartesian product.
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Dimensional Results for Cartesian Products of Homogeneous Moran Sets 被引量:1
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作者 Li CAO Xing-Gang HE 《Acta Mathematicae Applicatae Sinica》 SCIE CSCD 2012年第4期673-680,共8页
M(J, {ms * ns}, {Cs}) be the collection of Cartesian products of two homogenous Moran sets with the same ratios {cs} Where J = [0, 1] × [0, 1]. Then the maximal and minimal values of the Hausdorff dimensions f... M(J, {ms * ns}, {Cs}) be the collection of Cartesian products of two homogenous Moran sets with the same ratios {cs} Where J = [0, 1] × [0, 1]. Then the maximal and minimal values of the Hausdorff dimensions for the elements in M are obtained without any restriction on {msns} or {cs}. 展开更多
关键词 Homogeneous Moran sets cartesian product Hausdorff dimension
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Paired Domination of Cartesian Products of Graphs
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作者 Xin Min HOU Fan JIANG 《Journal of Mathematical Research and Exposition》 CSCD 2010年第1期181-185,共5页
Let γpr(G) denote the paired domination number and G □ H denote the Cartesian product of graphs G and H. In this paper we show that for all graphs G and H without isolated vertex, γpr(G)γpr(H)≤ 7γpr (G ... Let γpr(G) denote the paired domination number and G □ H denote the Cartesian product of graphs G and H. In this paper we show that for all graphs G and H without isolated vertex, γpr(G)γpr(H)≤ 7γpr (G □H). 展开更多
关键词 DOMINATION paired domination cartesian product.
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The g-Good-Neighbor Connectivity of Some Cartesian Product Graphs 被引量:1
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作者 Yinkui Li Ting Xie Xiaoxiao Qin 《Open Journal of Discrete Mathematics》 2023年第1期27-37,共11页
The g-good-neighbor connectivity of G is a generalization of the concept of connectivity, which is just for, and an important parameter in measuring the fault tolerance and reliability of interconnection network. Many... The g-good-neighbor connectivity of G is a generalization of the concept of connectivity, which is just for, and an important parameter in measuring the fault tolerance and reliability of interconnection network. Many well-known networks can be constructed by the Cartesian products of some simple graphs. In this paper, we determine the g-good-neighbor connectivity of some Cartesian product graphs. We give the exact value of g-good-neighbor connectivity of the Cartesian product of two complete graphs and for , mesh for , cylindrical grid and torus for . 展开更多
关键词 CONNECTIVITY The g-Good-Neighbor Connectivity cartesian Product
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Rupture Degree of Some Cartesian Product Graphs
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作者 Yinkui Li Tingyi Zhu 《Open Journal of Discrete Mathematics》 2023年第1期16-26,共11页
The rupture degree of a noncomplete-connected graph G is defined by , where is the number of components of and is the order of the largest component of. In this paper, we determine the rupture degree of some Cartesian... The rupture degree of a noncomplete-connected graph G is defined by , where is the number of components of and is the order of the largest component of. In this paper, we determine the rupture degree of some Cartesian product graphs. 展开更多
关键词 The Rupture Degree cartesian Product The Vulnerability
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The Path-Positive Property on the Products of Graphs
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作者 连广昌 《Journal of Southeast University(English Edition)》 EI CAS 1998年第2期130-134,共5页
The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. ... The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. It is proved that:① If the graphs G 1 and G 2 are the connected graphs, then the Cartesian product, the lexicographic product and the strong direct product in the products of graphs, are the path positive graphs. ② If the tensor product is a path positive graph if and only if the graph G 1 and G 2 are the connected graphs, and the graph G 1 or G 2 has an odd cycle and max{ λ 1μ 1,λ nμ m}≥2 in which λ 1 and λ n [ or μ 1 and μ m] are maximum and minimum characteristic values of graph G 1 [ or G 2 ], respectively. 展开更多
关键词 product of graphs path positive property cartesian product of graphs tensor product of graphs lexicographic product of graphs strong direct product of graphs
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Circular L(j,k)-labeling numbers of trees and products of graphs 被引量:3
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作者 吴琼 林文松 《Journal of Southeast University(English Edition)》 EI CAS 2010年第1期142-145,共4页
Let j, k and m be three positive integers, a circular m-L(j, k)-labeling of a graph G is a mapping f: V(G)→{0, 1, …, m-1}such that f(u)-f(v)m≥j if u and v are adjacent, and f(u)-f(v)m≥k if u and v are... Let j, k and m be three positive integers, a circular m-L(j, k)-labeling of a graph G is a mapping f: V(G)→{0, 1, …, m-1}such that f(u)-f(v)m≥j if u and v are adjacent, and f(u)-f(v)m≥k if u and v are at distance two,where a-bm=min{a-b,m-a-b}. The minimum m such that there exists a circular m-L(j, k)-labeling of G is called the circular L(j, k)-labeling number of G and is denoted by σj, k(G). For any two positive integers j and k with j≤k,the circular L(j, k)-labeling numbers of trees, the Cartesian product and the direct product of two complete graphs are determined. 展开更多
关键词 circular L(j k)-labeling number TREE cartesian product of graphs direct product of graphs
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Cartesian product over interval valued intuitionistic fuzzy sets 被引量:1
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作者 Jianming Xie Sanyang Liu 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2017年第6期1152-1161,共10页
The intuitionistic fuzzy set(IFS) based on fuzzy theory,which is of high efficiency to solve the fuzzy problem, has been introduced by Atanassov. Subsequently, he pushed the research one step further from the IFS to t... The intuitionistic fuzzy set(IFS) based on fuzzy theory,which is of high efficiency to solve the fuzzy problem, has been introduced by Atanassov. Subsequently, he pushed the research one step further from the IFS to the interval valued intuitionistic fuzzy set(IVIFS). On the basis of fuzzy set(FS), the IFS is a generalization concept. And the IFS is generalized to the IVIFS.In this paper, the definition of the sixth Cartesian product over IVIFSs is first introduced and its some properties are explored.We prove some equalities based on the operation and the relation over IVIFSs. Finally, we present one geometric interpretation and a numerical example of the sixth Cartesian product over IVIFSs. 展开更多
关键词 intuitionistic fuzzy sets(IFS) cartesian product ope ration geometric interpretation interval valued intuitionistic fuzzy set(IVIFS)
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Cartesian Product Based Transfer Learning Implementation for Brain Tumor Classification
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作者 Irfan Ahmed Usmani Muhammad Tahir Qadri +2 位作者 Razia Zia Asif Aziz Farheen Saeed 《Computers, Materials & Continua》 SCIE EI 2022年第11期4369-4392,共24页
Knowledge-based transfer learning techniques have shown good performance for brain tumor classification,especially with small datasets.However,to obtain an optimized model for targeted brain tumor classification,it is... Knowledge-based transfer learning techniques have shown good performance for brain tumor classification,especially with small datasets.However,to obtain an optimized model for targeted brain tumor classification,it is challenging to select a pre-trained deep learning(DL)model,optimal values of hyperparameters,and optimization algorithm(solver).This paper first presents a brief review of recent literature related to brain tumor classification.Secondly,a robust framework for implementing the transfer learning technique is proposed.In the proposed framework,a Cartesian product matrix is generated to determine the optimal values of the two important hyperparameters:batch size and learning rate.An extensive exercise consisting of 435 simulations for 11 state-of-the-art pre-trained DL models was performed using 16 paired hyperparameters from the Cartesian product matrix to input the model with the three most popular solvers(stochastic gradient descent with momentum(SGDM),adaptive moment estimation(ADAM),and root mean squared propagation(RMSProp)).The 16 pairs were formed using individual hyperparameter values taken from literature,which generally addressed only one hyperparameter for optimization,rather than making a grid for a particular range.The proposed framework was assessed using a multi-class publicly available dataset consisting of glioma,meningioma,and pituitary tumors.Performance assessment shows that ResNet18 outperforms all other models in terms of accuracy,precision,specificity,and recall(sensitivity).The results are also compared with existing state-of-the-art research work that used the same dataset.The comparison was mainly based on performance metric“accuracy”with support of three other parameters“precision,”“recall,”and“specificity.”The comparison shows that the transfer learning technique,implemented through our proposed framework for brain tumor classification,outperformed all existing approaches.To the best of our knowledge,the proposed framework is an efficient framework that helped reduce the computational complexity and the time to attain optimal values of two important hyperparameters and consequently the optimized model with an accuracy of 99.56%. 展开更多
关键词 Deep transfer learning cartesian product hyperparameter optimization magnetic resonance imaging(MRI) brain tumor classification
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Embedding Cartesian Product of Some Graphs in Books
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作者 YANG JIAO SHAO ZE-LING LI ZHI-GUO 《Communications in Mathematical Research》 CSCD 2018年第3期253-260,共8页
The book embedding of a graph G consists of placing the vertices of G in a line called spine and assigning edges of the graph to pages so that the edges assigned to the same page do not intersect. The number of pages ... The book embedding of a graph G consists of placing the vertices of G in a line called spine and assigning edges of the graph to pages so that the edges assigned to the same page do not intersect. The number of pages is the minimum number in which the graph can be embedded. In this paper, we study the book embedding of the Cartesian product Pm × Sn, Pm × Wn, Cn × Sm, Cn × Wm, and get an upper bound of their pagenumber. 展开更多
关键词 book embedding cartesian product pagenumber
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Geodetic Number and Geo-Chromatic Number of 2-Cartesian Product of Some Graphs
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作者 Medha Itagi Huilgol B. Divya 《Open Journal of Discrete Mathematics》 2022年第1期1-16,共16页
A set <em>S ⊆ V (G)</em> is called a geodetic set if every vertex of <em>G</em> lies on a shortest <em>u-v</em> path for some <em>u, v ∈ S</em>, the minimum cardinality... A set <em>S ⊆ V (G)</em> is called a geodetic set if every vertex of <em>G</em> lies on a shortest <em>u-v</em> path for some <em>u, v ∈ S</em>, the minimum cardinality among all geodetic sets is called geodetic number and is denoted by <img src="Edit_82259359-0135-4a65-9378-b767f0405b48.png" alt="" />. A set <em>C ⊆ V (G)</em> is called a chromatic set if <em>C</em> contains all vertices of different colors in<em> G</em>, the minimum cardinality among all chromatic sets is called the chromatic number and is denoted by <img src="Edit_d849148d-5778-459b-abbb-ff25b5cd659b.png" alt="" />. A geo-chromatic set<em> S</em><sub><em>c</em></sub><em> ⊆ V (G</em><em>)</em> is both a geodetic set and a chromatic set. The geo-chromatic number <img src="Edit_505e203c-888c-471c-852d-4b9c2dd1a31c.png" alt="" /><em> </em>of<em> G</em> is the minimum cardinality among all geo-chromatic sets of<em> G</em>. In this paper, we determine the geodetic number and the geo-chromatic number of 2-cartesian product of some standard graphs like complete graphs, cycles and paths. 展开更多
关键词 cartesian Product Grid Graphs Geodetic Set Geodetic Number Chromatic Set Chromatic Number Geo-Chromatic Set Geo-Chromatic Number
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L(s,t) edge spans of trees and product of two paths 被引量:1
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作者 牛庆杰 林文松 宋增民 《Journal of Southeast University(English Edition)》 EI CAS 2007年第4期639-642,共4页
L( s, t)-labeling is a variation of graph coloring which is motivated by a special kind of the channel assignment problem. Let s and t be any two nonnegative integers. An L (s, t)-labeling of a graph G is an assig... L( s, t)-labeling is a variation of graph coloring which is motivated by a special kind of the channel assignment problem. Let s and t be any two nonnegative integers. An L (s, t)-labeling of a graph G is an assignment of integers to the vertices of G such that adjacent vertices receive integers which differ by at least s, and vertices that are at distance of two receive integers which differ by at least t. Given an L(s, t) -labeling f of a graph G, the L(s, t) edge span of f, βst ( G, f) = max { |f(u) -f(v)|: ( u, v) ∈ E(G) } is defined. The L( s, t) edge span of G, βst(G), is minβst(G,f), where the minimum runs over all L(s, t)-labelings f of G. Let T be any tree with a maximum degree of △≥2. It is proved that if 2s≥t≥0, then βst(T) =( [△/2 ] - 1)t +s; if 0≤2s 〈 t and △ is even, then βst(T) = [ (△ - 1) t/2 ] ; and if 0 ≤2s 〈 t and △ is odd, then βst(T) = (△ - 1) t/2 + s. Thus, the L(s, t) edge spans of the Cartesian product of two paths and of the square lattice are completely determined. 展开更多
关键词 L(s t) -labeling L(s t) edge span TREE cartesian product square lattice
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L( 1,2)-edge- labeling for necklaces 被引量:1
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作者 贺丹 林文松 《Journal of Southeast University(English Edition)》 EI CAS 2014年第4期550-554,共5页
For a graph G and two positive integers j and k an m-L j k -edge-labeling of G is an assignment from the set 0 1 … m-to the edges such that adjacent edges receive labels that differ by at least j and edges at distanc... For a graph G and two positive integers j and k an m-L j k -edge-labeling of G is an assignment from the set 0 1 … m-to the edges such that adjacent edges receive labels that differ by at least j and edges at distance two receive labels that differ by at least k.Theλ′j k-number of G denoted byλ′j k G is the minimum integer m overall m-L j k -edge-labeling of G.The necklace is a specific type of Halin graph.The L 1 2 -edge-labeling of necklaces is studied and the lower and upper bounds on λ′1 2-number for necklaces are given.Also both the lower and upper bounds are attainable. 展开更多
关键词 channel assignment L j k -edge-labeling cartesian product Halin graph NECKLACE
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The Crossing Numbers of Cartesian-Products of Stars and 5-Vertex Graphs 被引量:2
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作者 HE Pei Ling QIAN Chun Hua +1 位作者 OUYANG Zhang Dong HUANG Yuan Qiu 《Journal of Mathematical Research and Exposition》 CSCD 2009年第2期335-342,共8页
In this paper, the crossing numbers of the Cartesian products of a specific 5-vertex graph with a star are given, and thus the result fills up the crossing number list of Cartesian products of all 5-vertex graphs with... In this paper, the crossing numbers of the Cartesian products of a specific 5-vertex graph with a star are given, and thus the result fills up the crossing number list of Cartesian products of all 5-vertex graphs with stars (presented by Marian Klesc). In addition, we also give an up to date description of Cartesian products of 5-vertex graphs with stars, whose crossing numbers are known. 展开更多
关键词 GRAPH DRAWING crossing number STAR cartesian product
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Graph Laplacian Matrix Learning from Smooth Time-Vertex Signal 被引量:1
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作者 Ran Li Junyi Wang +2 位作者 Wenjun Xu Jiming Lin Hongbing Qiu 《China Communications》 SCIE CSCD 2021年第3期187-204,共18页
In this paper,we focus on inferring graph Laplacian matrix from the spatiotemporal signal which is defined as“time-vertex signal”.To realize this,we first represent the signals on a joint graph which is the Cartesia... In this paper,we focus on inferring graph Laplacian matrix from the spatiotemporal signal which is defined as“time-vertex signal”.To realize this,we first represent the signals on a joint graph which is the Cartesian product graph of the time-and vertex-graphs.By assuming the signals follow a Gaussian prior distribution on the joint graph,a meaningful representation that promotes the smoothness property of the joint graph signal is derived.Furthermore,by decoupling the joint graph,the graph learning framework is formulated as a joint optimization problem which includes signal denoising,timeand vertex-graphs learning together.Specifically,two algorithms are proposed to solve the optimization problem,where the discrete second-order difference operator with reversed sign(DSODO)in the time domain is used as the time-graph Laplacian operator to recover the signal and infer a vertex-graph in the first algorithm,and the time-graph,as well as the vertex-graph,is estimated by the other algorithm.Experiments on both synthetic and real-world datasets demonstrate that the proposed algorithms can effectively infer meaningful time-and vertex-graphs from noisy and incomplete data. 展开更多
关键词 cartesian product graph discrete secondorder difference operator Gaussian prior distribution graph Laplacian matrix learning spatiotemporal smoothness time-vertex signal
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New Concepts on Quadripartitioned Bipolar Single Valued Neutrosophic Graph 被引量:1
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作者 S.Satham Hussain G.Muhiuddin +1 位作者 N.Durga D.Al-Kadi 《Computer Modeling in Engineering & Sciences》 SCIE EI 2022年第1期559-580,共22页
The partition of indeterminacy function of the neutrosophic set into the contradiction part and the ignorance part represent the quadripartitioned single valued neutrosophic set.In this work,the new concept of quadrip... The partition of indeterminacy function of the neutrosophic set into the contradiction part and the ignorance part represent the quadripartitioned single valued neutrosophic set.In this work,the new concept of quadripartitioned bipolar single valued neutrosophic graph is established,and the operations on it are studied.The Cartesian product,cross product,lexicographic product,strong product and composition of quadripartitioned bipolar single valued neutrosophic graph are investigated.The proposed concepts are illustrated with examples. 展开更多
关键词 Quadripartitioned bipolar single valued neutrosophic graph operations of quadripartitioned bipolar single valued neutrosophic graph cartesian product
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