Let R be a commutative ring and U(R)the multiplicative group of unit elements of R.In 2012,Khashyarmanesh et al.defined the generalized unit and unitary Cayley graph,T(R,G,S),corresponding to a multiplicative subgroup...Let R be a commutative ring and U(R)the multiplicative group of unit elements of R.In 2012,Khashyarmanesh et al.defined the generalized unit and unitary Cayley graph,T(R,G,S),corresponding to a multiplicative subgroup G of U(R)and a nonempty subset S of G with S^(-1)={s^(-1)|s∈S}■S,asthegraphwithvertexsetR and two distinct vertices x and y being adjacent if and only if there exists s∈S such that x+sy∈G.In this paper,we characterize all Artinian rings R for which T(R,U(R),S)is projective.This leads us to determine all Artinian rings whose unit graphs,unitary Cayley graphs and co-maximal graphs are projective.In addition,we prove that for an Artinian ring R for which T(R,U(R),S)has finite nonorientable genus,R must be a finite ring.Finally,it is proved that for a given positive integer k,the number of finite rings R for which T(R,U(R),S)has nonorientable genus k is finite.展开更多
In this paper, we show that the nonorientable genus of Cm ＋ Cn, the join of two cycles Cm and Cn, is equal to [（（m-2）（n-2））/2] if m = 3, n ≡ 1 （mod 2）, or m ≥ 4, n ≥ 4, （m, n） （4, 4）. We determine that...In this paper, we show that the nonorientable genus of Cm ＋ Cn, the join of two cycles Cm and Cn, is equal to [（（m-2）（n-2））/2] if m = 3, n ≡ 1 （mod 2）, or m ≥ 4, n ≥ 4, （m, n） （4, 4）. We determine that the nonorientable genus of C4 ＋C4 is 3, and that the nonorientable genus of C3 ＋Cn is n/2 if n ≡ 0 （mod 2）. Our results show that a minimum nonorientable genus embedding of the complete bipartite graph Km,n cannot be extended to an embedding of the join of two cycles without increasing the genus of the surface.展开更多
In this paper,let m≥1 be an integer,M be an m-dimensional compact Riemannian manifold.Firstly the linearized Poincare map of the Lagrangian system at critical point x d/dt L_(q)(t,x,x)−L_(p)(t,x,x)=0 is explicitly gi...In this paper,let m≥1 be an integer,M be an m-dimensional compact Riemannian manifold.Firstly the linearized Poincare map of the Lagrangian system at critical point x d/dt L_(q)(t,x,x)−L_(p)(t,x,x)=0 is explicitly given,then we prove that Morse index and Maslov-type index of x are well defined whether the manifold M is orientable or not via the parallel transport method which makes no appeal to unitary trivialization and establish the relation of Morse index and Maslov-type index,finally derive a criterion for the instability of critical point and orientation of M and obtain the formula for two Maslov-type indices.展开更多
文摘Let R be a commutative ring and U(R)the multiplicative group of unit elements of R.In 2012,Khashyarmanesh et al.defined the generalized unit and unitary Cayley graph,T(R,G,S),corresponding to a multiplicative subgroup G of U(R)and a nonempty subset S of G with S^(-1)={s^(-1)|s∈S}■S,asthegraphwithvertexsetR and two distinct vertices x and y being adjacent if and only if there exists s∈S such that x+sy∈G.In this paper,we characterize all Artinian rings R for which T(R,U(R),S)is projective.This leads us to determine all Artinian rings whose unit graphs,unitary Cayley graphs and co-maximal graphs are projective.In addition,we prove that for an Artinian ring R for which T(R,U(R),S)has finite nonorientable genus,R must be a finite ring.Finally,it is proved that for a given positive integer k,the number of finite rings R for which T(R,U(R),S)has nonorientable genus k is finite.
基金Supported by National Natural Science Foundation of China(Grant No.11171114)
文摘In this paper, we show that the nonorientable genus of Cm ＋ Cn, the join of two cycles Cm and Cn, is equal to [（（m-2）（n-2））/2] if m = 3, n ≡ 1 （mod 2）, or m ≥ 4, n ≥ 4, （m, n） （4, 4）. We determine that the nonorientable genus of C4 ＋C4 is 3, and that the nonorientable genus of C3 ＋Cn is n/2 if n ≡ 0 （mod 2）. Our results show that a minimum nonorientable genus embedding of the complete bipartite graph Km,n cannot be extended to an embedding of the join of two cycles without increasing the genus of the surface.
文摘In this paper,let m≥1 be an integer,M be an m-dimensional compact Riemannian manifold.Firstly the linearized Poincare map of the Lagrangian system at critical point x d/dt L_(q)(t,x,x)−L_(p)(t,x,x)=0 is explicitly given,then we prove that Morse index and Maslov-type index of x are well defined whether the manifold M is orientable or not via the parallel transport method which makes no appeal to unitary trivialization and establish the relation of Morse index and Maslov-type index,finally derive a criterion for the instability of critical point and orientation of M and obtain the formula for two Maslov-type indices.
