Dear Editor,This letter proposes the graph tensor alliance attention network(GT-A^(2)T)to represent a dynamic graph(DG)precisely.Its main idea includes 1)Establishing a unified spatio-temporal message propagation fram...Dear Editor,This letter proposes the graph tensor alliance attention network(GT-A^(2)T)to represent a dynamic graph(DG)precisely.Its main idea includes 1)Establishing a unified spatio-temporal message propagation framework on a DG via the tensor product for capturing the complex cohesive spatio-temporal interdependencies precisely and 2)Acquiring the alliance attention scores by node features and favorable high-order structural correlations.展开更多
This paper investigates the robust graph coloring problem with application to a kind of examination timetabling by using the matrix semi-tensor product, and presents a number of new results and algorithms. First, usin...This paper investigates the robust graph coloring problem with application to a kind of examination timetabling by using the matrix semi-tensor product, and presents a number of new results and algorithms. First, using the matrix semi-tensor product, the robust graph coloring is expressed into a kind of optimization problem taking in an algebraic form of matrices, based on which an algorithm is designed to find all the most robust coloring schemes for any simple graph. Second, an equivalent problem of robust graph coloring is studied, and a necessary and sufficient condition is proposed, from which a new algorithm to find all the most robust coloring schemes is established. Third, a kind of examination timetabling is discussed by using the obtained results, and a method to design a practicable timetabling scheme is presented. Finally, the effectiveness of the results/algorithms presented in this paper is shown by two illustrative examples.展开更多
This paper investigates the connections between ring theory, module theory, and graph theory through the graph G(R)of a ring R. We establish that vertices of G(R)correspond to modules, with edges defined by the vanish...This paper investigates the connections between ring theory, module theory, and graph theory through the graph G(R)of a ring R. We establish that vertices of G(R)correspond to modules, with edges defined by the vanishing of their tensor product. Key results include the graph’s connectivity, a diameter of at most 3, and a girth of at most 7 when cycles are present. We show that the set of modules S(R)is empty if and only if R is a field, and that for semisimple rings, the diameter is at most 2. The paper also discusses module isomorphisms over subrings and localization, as well as the inclusion of G(T)within G(R)for a quotient ring T, highlighting that the reverse inclusion is not guaranteed. Finally, we provide an example illustrating that a non-finitely generated module M does not imply M⊗M=0. These findings deepen our understanding of the interplay among rings, modules, and graphs.展开更多
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
This article is devoted to developing a deep learning method for the numerical solution of the partial differential equations (PDEs). Graph kernel neural networks (GKNN) approach to embedding graphs into a computation...This article is devoted to developing a deep learning method for the numerical solution of the partial differential equations (PDEs). Graph kernel neural networks (GKNN) approach to embedding graphs into a computationally numerical format has been used. In particular, for investigation mathematical models of the dynamical system of cancer cell invasion in inhomogeneous areas of human tissues have been considered. Neural operators were initially proposed to model the differential operator of PDEs. The GKNN mapping features between input data to the PDEs and their solutions have been constructed. The boundary integral method in combination with Green’s functions for a large number of boundary conditions is used. The tools applied in this development are based on the Fourier neural operators (FNOs), graph theory, theory elasticity, and singular integral equations.展开更多
基金supported in part by the National Natural Science Foundation of China(62372385).
文摘Dear Editor,This letter proposes the graph tensor alliance attention network(GT-A^(2)T)to represent a dynamic graph(DG)precisely.Its main idea includes 1)Establishing a unified spatio-temporal message propagation framework on a DG via the tensor product for capturing the complex cohesive spatio-temporal interdependencies precisely and 2)Acquiring the alliance attention scores by node features and favorable high-order structural correlations.
基金This work was supported by the National Natural Science Foundation of China （Nos. G61374065, G61034007, G61374002） the Fund for the Taishan Scholar Project of Shandong Province, the Natural Science Foundation of Shandong Province （No. ZR2010FM013） the Scientific Research and Development Project of Shandong Provincial Education Department （No. J11LA01 ）
文摘This paper investigates the robust graph coloring problem with application to a kind of examination timetabling by using the matrix semi-tensor product, and presents a number of new results and algorithms. First, using the matrix semi-tensor product, the robust graph coloring is expressed into a kind of optimization problem taking in an algebraic form of matrices, based on which an algorithm is designed to find all the most robust coloring schemes for any simple graph. Second, an equivalent problem of robust graph coloring is studied, and a necessary and sufficient condition is proposed, from which a new algorithm to find all the most robust coloring schemes is established. Third, a kind of examination timetabling is discussed by using the obtained results, and a method to design a practicable timetabling scheme is presented. Finally, the effectiveness of the results/algorithms presented in this paper is shown by two illustrative examples.
文摘This paper investigates the connections between ring theory, module theory, and graph theory through the graph G(R)of a ring R. We establish that vertices of G(R)correspond to modules, with edges defined by the vanishing of their tensor product. Key results include the graph’s connectivity, a diameter of at most 3, and a girth of at most 7 when cycles are present. We show that the set of modules S(R)is empty if and only if R is a field, and that for semisimple rings, the diameter is at most 2. The paper also discusses module isomorphisms over subrings and localization, as well as the inclusion of G(T)within G(R)for a quotient ring T, highlighting that the reverse inclusion is not guaranteed. Finally, we provide an example illustrating that a non-finitely generated module M does not imply M⊗M=0. These findings deepen our understanding of the interplay among rings, modules, and graphs.
文摘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.
文摘This article is devoted to developing a deep learning method for the numerical solution of the partial differential equations (PDEs). Graph kernel neural networks (GKNN) approach to embedding graphs into a computationally numerical format has been used. In particular, for investigation mathematical models of the dynamical system of cancer cell invasion in inhomogeneous areas of human tissues have been considered. Neural operators were initially proposed to model the differential operator of PDEs. The GKNN mapping features between input data to the PDEs and their solutions have been constructed. The boundary integral method in combination with Green’s functions for a large number of boundary conditions is used. The tools applied in this development are based on the Fourier neural operators (FNOs), graph theory, theory elasticity, and singular integral equations.
