The metric dimension problem is called navigation problem due to its application to robot navigation in space.Further this concept has wide applications in motion planning,sonar and loran station,and so on.In this pap...The metric dimension problem is called navigation problem due to its application to robot navigation in space.Further this concept has wide applications in motion planning,sonar and loran station,and so on.In this paper,we study certain results on the metric dimension,upper dimension and resolving number of extended annihilating-ideal graph EAG(R)associated to a commutative ring R,denoted by dim M(EAG(R)),dim+(EAG(R))and res(EAG(R)),respectively.Here we prove the finiteness conditions of dim M(EAG(R))and dim+(EAG(R)).In addition,we characterize dim M(EAG(R)),dim+(EAG(R))and res(EAG(R))for artinian rings and the direct product of rings.展开更多
Let R be a commutative ring and A（R） be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG（R） with the vertex set A（R）＊ = A（R）／{（0）} and two distinct...Let R be a commutative ring and A（R） be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG（R） with the vertex set A（R）＊ = A（R）／{（0）} and two distinct vertices I and J are adjacent if and only if IJ = （0）. Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and w（AG（R）） = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.展开更多
In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete char- acterization is established for a finite local ...In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete char- acterization is established for a finite local ring whose annihilating-ideal graph is a star graph.展开更多
文摘The metric dimension problem is called navigation problem due to its application to robot navigation in space.Further this concept has wide applications in motion planning,sonar and loran station,and so on.In this paper,we study certain results on the metric dimension,upper dimension and resolving number of extended annihilating-ideal graph EAG(R)associated to a commutative ring R,denoted by dim M(EAG(R)),dim+(EAG(R))and res(EAG(R)),respectively.Here we prove the finiteness conditions of dim M(EAG(R))and dim+(EAG(R)).In addition,we characterize dim M(EAG(R)),dim+(EAG(R))and res(EAG(R))for artinian rings and the direct product of rings.
文摘Let R be a commutative ring and A（R） be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph AG（R） with the vertex set A（R）＊ = A（R）／{（0）} and two distinct vertices I and J are adjacent if and only if IJ = （0）. Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and w（AG（R）） = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.
基金The first author is supported by Fundamental Research Funds for the Central Universi- ties （No. XDJK2013C060）, Chongqing Research Program of Application Foundation and Advanced Technology （No. cstc2014jcyjA00028） and Scientific Research Foundation for Doctors of Southwest University （No. SWUl12054）. The second author is supported by National Natural Science Foundation of China （No. 11271250）.
文摘In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete char- acterization is established for a finite local ring whose annihilating-ideal graph is a star graph.
