On Comaximal Graphs of Near-rings
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Let N be a zero-symmetric near-ring with identity and let Γ(N) be a graph with vertices as elements of N, where two different vertices a and b are adjacent if and only if hai + hbi = N, where hxi is the ideal of N generated by x. Let Γ1(N) be the subgraph of Γ(N) generated by the set {n ∈ N : hni = N} and Γ2(N) be the subgraph of Γ(N) generated by the set N＼v(Γ1(N)), where v(G) is the set of all vertices of a graph G. In this paper, we completely characterize the diameter of the subgraph Γ2(N) of Γ(N). In addition, it is shown that for any near-ring, Γ2(N)＼(N) is a complete bipartite graph if and only if the number of maximal ideals of N is 2, where M(N) is the intersection of all maximal ideals of N and Γ2(N)＼(N) is the graph obtained by removing the elements of the set M(N) from the vertices set of the graph Γ2(N).
Abstract
Let N be a zero-symmetric near-ring with identity and let Γ(N) be a graph with vertices as elements of N, where two different vertices a and b are adjacent if and only if hai + hbi = N, where hxi is the ideal of N generated by x. Let Γ1(N) be the subgraph of Γ(N) generated by the set {n ∈ N : hni = N} and Γ2(N) be the subgraph of Γ(N) generated by the set N＼v(Γ1(N)), where v(G) is the set of all vertices of a graph G. In this paper, we completely characterize the diameter of the subgraph Γ2(N) of Γ(N). In addition, it is shown that for any near-ring, Γ2(N)＼(N) is a complete bipartite graph if and only if the number of maximal ideals of N is 2, where M(N) is the intersection of all maximal ideals of N and Γ2(N)＼(N) is the graph obtained by removing the elements of the set M(N) from the vertices set of the graph Γ2(N).
참고문헌 (6)
- D. D. Anderson, "Beck’s coloring of a commutative ring", J. Algebra, 159, pp.500~514 (1993)
- I. Beck, "Coloring of commutative rings", J. Algebra, 116, pp.208~226 (1988)
- H. R. Maimani, "Comaximal graph of commutative rings", J. Algebra, 319(4), pp.1801~1808 (2008)
- G. Mason, "Reflexive ideals", Comm. Algebra, 9, pp.1709~1724 (1988)
- G. Pilz, "Near-Rings" (1983), North-Holland
- P. K. Sharma, "A note on graphical representation of rings", J. Algebra, 176, pp.124~127 (1995)