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Some classes of perfect strongly annihilating-ideal graphs associated with commutative rings

Mitra Jalali, Abolfazl Tehranian, Reza Nikandish, Hamid Rasouli (2020)

Commentationes Mathematicae Universitatis Carolinae

Let R be a commutative ring with identity and A ( R ) be the set of ideals with nonzero annihilator. The strongly annihilating-ideal graph of R is defined as the graph SAG ( R ) with the vertex set A ( R ) * = A ( R ) ∖ { 0 } and two distinct vertices I and J are adjacent if and only if I ∩ Ann ( J ) ≠ ( 0 ) and J ∩ Ann ( I ) ≠ ( 0 ) . In this paper, the perfectness of SAG ( R ) for some classes of rings R is investigated.

Some results on the annihilator graph of a commutative ring

Mojgan Afkhami, Kazem Khashyarmanesh, Zohreh Rajabi (2017)

Czechoslovak Mathematical Journal

Let R be a commutative ring. The annihilator graph of R , denoted by AG ( R ) , is the undirected graph with all nonzero zero-divisors of R as vertex set, and two distinct vertices x and y are adjacent if and only if ann R ( x y ) ≠ ann R ( x ) ∪ ann R ( y ) , where for z ∈ R , ann R ( z ) = { r ∈ R : r z = 0 } . In this paper, we characterize all finite commutative rings R with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings R whose annihilator graphs have clique number 1 , 2 or 3 . Also, we investigate some properties of the annihilator...

Stable graphs

Klaus-Peter Podewski, Martin Ziegler (1978)

Fundamenta Mathematicae

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