The Zero-divisor Graph of the Ring of Integers Modulo n
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Let Zn be the ring of integers modulo n and Γ(Zn) the zero-divisor graph of Zn. In this paper, we study some properties of Γ(Zn). More precisely, we completely characterize the diameter and the girth of Γ(Zn). We also calculate the chromatic number of Γ(Zn).E(G), deg(u) > deg(v)g}. The minimum cardinality of a strong (weak) ecient dominating set of G is called the strong (weak) ecient domination number of G and is denoted by se(G) (we(G)). A graph G is strong ecient if there exists a strong ecient dominating set of G. In this paper, some cycle and star related Nordhaus-Gaddum type relations on strong ecient dominating sets and the number of strong ecient dominating sets are studied.
Abstract
Let Zn be the ring of integers modulo n and Γ(Zn) the zero-divisor graph of Zn. In this paper, we study some properties of Γ(Zn). More precisely, we completely characterize the diameter and the girth of Γ(Zn). We also calculate the chromatic number of Γ(Zn).E(G), deg(u) > deg(v)g}. The minimum cardinality of a strong (weak) ecient dominating set of G is called the strong (weak) ecient domination number of G and is denoted by se(G) (we(G)). A graph G is strong ecient if there exists a strong ecient dominating set of G. In this paper, some cycle and star related Nordhaus-Gaddum type relations on strong ecient dominating sets and the number of strong ecient dominating sets are studied.
참고문헌 (5)
- D. F. Anderson, "Commutative Algebra: Noetherian and Non-Noetherian Perspectives", pp.23~45 (2011), Springer
- D. F. Anderson, "The zero-divisor graph of a commutative ring", J. Algebra, 217, pp.434~447 (1999)
- 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)
- S. Mulay, "Cycles and symmetries of zero-divisors", Comm. Algebra, 30, pp.3533~3558 (2002)