Let_R be a commutative ring with non-zero identity and I its proper ideal.The total graph of R with respect to I ,denoted by T(ГI(R)),is the undirected graph with all elements of R as vertices,and where distinct vert...Let_R be a commutative ring with non-zero identity and I its proper ideal.The total graph of R with respect to I ,denoted by T(ГI(R)),is the undirected graph with all elements of R as vertices,and where distinct vert ices x and y are adjacen t if and only if x+y∈S(I)={a ∈ R:ra ∈ I for some r∈B\I}.In this paper,some bounds for the genus of T(ГI(R)) are obtained.We improve and generalize some results for the total graphs of commutative rings.In addition,we obtain an isomorphism relation between two ideal based total graphs.展开更多
Let R be a ring with non-zero identity.The unit graph of R,denoted by G(R),is an undirected graph with all the elements of R as vertices and where distinct vertices x,y are adjacent if and only if x+y is a unit of R.I...Let R be a ring with non-zero identity.The unit graph of R,denoted by G(R),is an undirected graph with all the elements of R as vertices and where distinct vertices x,y are adjacent if and only if x+y is a unit of R.In this paper,we investigate the Wiener index and hyper-Wiener index of G(R)and explicitly determine their values.展开更多
文摘Let_R be a commutative ring with non-zero identity and I its proper ideal.The total graph of R with respect to I ,denoted by T(ГI(R)),is the undirected graph with all elements of R as vertices,and where distinct vert ices x and y are adjacen t if and only if x+y∈S(I)={a ∈ R:ra ∈ I for some r∈B\I}.In this paper,some bounds for the genus of T(ГI(R)) are obtained.We improve and generalize some results for the total graphs of commutative rings.In addition,we obtain an isomorphism relation between two ideal based total graphs.
基金supported in part by a grant from Science and Engineering Research Board,India(Ref.MTR/2017/000830)supported by the National Natural Science Foundation of China(Ref.11961050)the Guangxi Natural Sciences Foundation(Ref.2021GXNSFAA220043).
文摘Let R be a ring with non-zero identity.The unit graph of R,denoted by G(R),is an undirected graph with all the elements of R as vertices and where distinct vertices x,y are adjacent if and only if x+y is a unit of R.In this paper,we investigate the Wiener index and hyper-Wiener index of G(R)and explicitly determine their values.
