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On the Nonorientable Genus of the Generalized Unit and Unitary Cayley Graphs of a Commutative Ring
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作者 Mahdi Reza Khorsandi Seyed Reza Musawi 《Algebra Colloquium》 SCIE CSCD 2022年第1期167-180,共14页
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
关键词 unit graph unitary Cayley graph co-maximal graph projective graph nonorientable genus
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Embedding generalized of circulant graphs and Petersen graphs on projective plane
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作者 Yah YANG Yanpei LIU 《Frontiers of Mathematics in China》 SCIE CSCD 2015年第1期209-220,共12页
Both the circulant graph and the generalized Petersen graph are important types of graphs in graph theory. In this paper, the structures of embeddings of circulant graph C(2n + 1; {1, n}) on the projective plane ar... Both the circulant graph and the generalized Petersen graph are important types of graphs in graph theory. In this paper, the structures of embeddings of circulant graph C(2n + 1; {1, n}) on the projective plane are described, the number of embeddings of C(2n + 1; {1, n}) on the projective plane follows, then the number of embeddings of the generalized Petersen graph P(2n + 1, n) on the projective plane is deduced from that of C(2n + 1; {1, n}), because C(2n + 1; {1, n}) is a minor of P(2n + 1, n), their structures of embeddings have relations. In the same way, the number of embeddings of the generalized Petersen graph P(2n, 2) on the projective plane is also obtained. 展开更多
关键词 EMBEDDING joint tree circulant graph generalized Petersen graph projective plane
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