Given two ideals I and J of a commutative ring R,there are two extreme connections between I and J:I+J=R and I∩J={0}.For the former case,graphs whose vertices are defined as the proper ideals of R and that two vertic...Given two ideals I and J of a commutative ring R,there are two extreme connections between I and J:I+J=R and I∩J={0}.For the former case,graphs whose vertices are defined as the proper ideals of R and that two vertices are adjacent if and only if their sum is the whole ring R are known as co-maximal ideal graphs.In this paper,we introduce a new kind of graph structure on R,called co-minimal ideal graph,according to the second case:Its vertices are the nonzero ideals of R and two vertices are adjacent if and only if their intersection is zero.Some important graph parameters(including girth,diameter,clique number and chromatic number)and graph structures(including tree and bipartite graph)of co-minimal ideal graphs over finite commutative rings are studied.In particular,we show that the co-maximal ideal graph and the co-minimal ideal graph over R are isomorphic if and only if the number of maximal ideals of R and the number of minimal ideals of R coincide.展开更多
基金partially supported by the Open Research Fund of Key Laboratory of Nonlinear Analysis&Applications(Central China Normal University),Ministry of Education,P.R.Chinathe Guiding Science and Technology Plan Project of Suqian City in 2023(No.Z2023130)partially supported by NSFC(No.12271234)。
文摘Given two ideals I and J of a commutative ring R,there are two extreme connections between I and J:I+J=R and I∩J={0}.For the former case,graphs whose vertices are defined as the proper ideals of R and that two vertices are adjacent if and only if their sum is the whole ring R are known as co-maximal ideal graphs.In this paper,we introduce a new kind of graph structure on R,called co-minimal ideal graph,according to the second case:Its vertices are the nonzero ideals of R and two vertices are adjacent if and only if their intersection is zero.Some important graph parameters(including girth,diameter,clique number and chromatic number)and graph structures(including tree and bipartite graph)of co-minimal ideal graphs over finite commutative rings are studied.In particular,we show that the co-maximal ideal graph and the co-minimal ideal graph over R are isomorphic if and only if the number of maximal ideals of R and the number of minimal ideals of R coincide.
