For a finite group G,the co-maximal subgroup graphΓ(G)of G is a graph whose vertices are proper subgroups of G,and two distinct vertices H and K are adjacent if and only if H K=G.The deleted co-maximal subgroup graph...For a finite group G,the co-maximal subgroup graphΓ(G)of G is a graph whose vertices are proper subgroups of G,and two distinct vertices H and K are adjacent if and only if H K=G.The deleted co-maximal subgroup graphΓ^(∗)(G)is obtained by removing isolated vertices fromΓ(G).Firstly,we provide necessary and sufficient conditions forΓ^(∗)(G)to be connected;in particular,from the viewpoint of normal subgroups in G,we give some sufficient conditions forΓ^(∗)(G)to be connected.Secondly,for a finite abelian group G we prove that the diameter ofΓ^(∗)(G),diam(Γ^(∗)(G)),is at most 3.Also,we characterize G with diam(Γ^(∗)(G))=i for i=1,2,3 and we give characterizations for G withΓ^(∗)(G)being complete bipartite graphs and null graphs separately.Finally,we show that for the semidirect product G of two finite cyclic groups,Γ^(∗)(G)is connected and diam(Γ^(∗)(G))=2.展开更多
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
基金supported by National Natural Science Foundation of China(Grant Nos.12071194,12361072)2023 Xinjiang Uygur Autonomous Region Natural Science Foundation General Project(No.2023D01A36)2023 Xinjiang Uygur Autonomous Region Natural Science Foundation for Youths(No.2023D01B48).
文摘For a finite group G,the co-maximal subgroup graphΓ(G)of G is a graph whose vertices are proper subgroups of G,and two distinct vertices H and K are adjacent if and only if H K=G.The deleted co-maximal subgroup graphΓ^(∗)(G)is obtained by removing isolated vertices fromΓ(G).Firstly,we provide necessary and sufficient conditions forΓ^(∗)(G)to be connected;in particular,from the viewpoint of normal subgroups in G,we give some sufficient conditions forΓ^(∗)(G)to be connected.Secondly,for a finite abelian group G we prove that the diameter ofΓ^(∗)(G),diam(Γ^(∗)(G)),is at most 3.Also,we characterize G with diam(Γ^(∗)(G))=i for i=1,2,3 and we give characterizations for G withΓ^(∗)(G)being complete bipartite graphs and null graphs separately.Finally,we show that for the semidirect product G of two finite cyclic groups,Γ^(∗)(G)is connected and diam(Γ^(∗)(G))=2.
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.
