In this paper, both the roman domination number and the number of minimum roman dominating sets are found for any rectangular rook’s graph. In a similar fashion, the roman domination number and the number of minimum ...In this paper, both the roman domination number and the number of minimum roman dominating sets are found for any rectangular rook’s graph. In a similar fashion, the roman domination number and the number of minimum roman dominating sets are found on the square bishop’s graph for odd board sizes. Also found are the number of minimum total dominating sets associated with the light-colored squares when n?≡1(mod12)? (with n>1), and same for the dark-colored squares when n?≡7(mod12) .展开更多
Let X and Y be Banach spaces.For A∈L(X),B∈L(Y),C∈L(Y,X),let MCbe the operator matrix defined on X⊕Y by M_(C)=(AC0B)∈L(X⊕Y).In this paper we investigate the decomposability for MC.We consider Bishop’s property(...Let X and Y be Banach spaces.For A∈L(X),B∈L(Y),C∈L(Y,X),let MCbe the operator matrix defined on X⊕Y by M_(C)=(AC0B)∈L(X⊕Y).In this paper we investigate the decomposability for MC.We consider Bishop’s property(β),decomposition property(δ)and Dunford’s property(C)and obtain the relationship of these properties between M_(C) and its entries.We explore how σ_(*)(M_(C))shrinks from σ_(*)(A)∪σ_(*)(B),where σ_(*)denotes σ_(β),σ_(δ),σ_(C),σ_(dec).In particular,we develop some sufficient conditions for equality σ_(*)(MC)=σ_(*)(A)∪σ_(*)(B).Besides,we consider the perturbation of these properties for MCand show that in perturbing with certain operators C the properties for MCkeeps with A,B.Some examples are given to illustrate our results.Furthermore,we study the decomposability for(0AB0).Finally,we give applications of decomposability for operator matrices.展开更多
文摘In this paper, both the roman domination number and the number of minimum roman dominating sets are found for any rectangular rook’s graph. In a similar fashion, the roman domination number and the number of minimum roman dominating sets are found on the square bishop’s graph for odd board sizes. Also found are the number of minimum total dominating sets associated with the light-colored squares when n?≡1(mod12)? (with n>1), and same for the dark-colored squares when n?≡7(mod12) .
基金Supported by National Natural Science Foundation of China(Grant No.11761029)Inner Mongolia Higher Education Science and Technology Research Project(Grant Nos.NJZY22323 and NJZY22324)Inner Mongolia Natural Science Foundation(Grant No.2018MS07020)。
文摘Let X and Y be Banach spaces.For A∈L(X),B∈L(Y),C∈L(Y,X),let MCbe the operator matrix defined on X⊕Y by M_(C)=(AC0B)∈L(X⊕Y).In this paper we investigate the decomposability for MC.We consider Bishop’s property(β),decomposition property(δ)and Dunford’s property(C)and obtain the relationship of these properties between M_(C) and its entries.We explore how σ_(*)(M_(C))shrinks from σ_(*)(A)∪σ_(*)(B),where σ_(*)denotes σ_(β),σ_(δ),σ_(C),σ_(dec).In particular,we develop some sufficient conditions for equality σ_(*)(MC)=σ_(*)(A)∪σ_(*)(B).Besides,we consider the perturbation of these properties for MCand show that in perturbing with certain operators C the properties for MCkeeps with A,B.Some examples are given to illustrate our results.Furthermore,we study the decomposability for(0AB0).Finally,we give applications of decomposability for operator matrices.
