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首页> 外文期刊>Journal of algebra and its applications >The intersection graph of gamma sets in the total graph of a commutative ring-II
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The intersection graph of gamma sets in the total graph of a commutative ring-II

机译：交换环II的总图中的伽玛集的交集图

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The intersection graph I_(TΓ)(R) of gamma sets in the total graph T_Γ(R) of a commutative ring R, is the undirected graph with vertex set as the collection of all γ-sets in the total graph of R and two distinct vertices u and v are adjacent if and only if u ∩ v ≠. Tamizh Chelvam and Asir [The intersection graph of gamma sets in the total graph I, to appear in J. Algebra Appl.] studied about I_(TΓ)(R) where R is a commutative Artin ring. In this paper, we continue our interest on I _(TΓ)(R) and actually we study about Eulerian, Hamiltonian and pancyclic nature of I_(TΓ)(R). Further, we focus on certain graph theoretic parameters of I_(TΓ)(R) like the independence number, the clique number and the connectivity of I_(TΓ)(R). Also, we obtain both vertex and edge chromatic numbers of I_(TΓ)(R). In fact, it is proved that if R is a finite commutative ring, then χ(I _(TΓ)(R)) = ω(I_(TΓ)(R)). Having proved that I_(TΓ)(R) is weakly perfect for all finite commutative rings, we further characterize all finite commutative rings for which I _(TΓ)(R) is perfect. In this sequel, we characterize all commutative Artin rings for which I_(TΓ)(R) is of class one (i.e. χ′(I_(TΓ)(R)) = Δ(I_(TΓ)(R))). Finally, it is proved that the vertex connectivity and edge connectivity of ITΓ(R) are equal to the degree of any vertex in I TΓ(R).
机译：交换环R的总图T_Γ（R）中的伽玛集的交点图I_（TΓ）（R）是无向图，其顶点集为R和两个总图中所有γ集的集合当且仅当u∩v≠时，u和v的相邻顶点相邻。 Tamizh Chelvam和Asir [γ在总图I中的交集，将出现在J. Algebra Appl。中]研究了I_（TΓ）（R），其中R是可交换的Artin环。在本文中，我们继续关注I_（TΓ）（R），实际上我们研究了I_（TΓ）（R）的欧拉，哈密顿和泛环性质。此外，我们关注I_（TΓ）（R）的某些图论参数，例如I_（TΓ）（R）的独立性，集团数和连通性。同样，我们获得I_（TΓ）（R）的顶点和边缘色数。实际上，证明了如果R是有限的交换环，则χ（I _（TΓ）（R））=ω（I_（TΓ）（R））。证明I_（TΓ）（R）对于所有有限的交换环都是弱理想的，我们进一步表征了I_（TΓ）（R）理想的所有有限交换环。在此续集中，我们表征所有I_（TΓ）（R）为一类（即χ'（I_（T _）（R））=Δ（I_（TΓ）（R）））的所有交换Artin环。最后，证明ITΓ（R）的顶点连通性和边缘连通性等于ITΓ（R）中任何顶点的程度。

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