Let R be a commutative ring, I(R) be the set of all ideals of R and S be a subset of I * (R) = I(... more Let R be a commutative ring, I(R) be the set of all ideals of R and S be a subset of I * (R) = I(R) \ {0}. We define a Cayley sum digraph of ideals of R, denoted by − − → Cay + (I(R), S), as a directed graph whose vertex set is the set I(R) and, for every two distinct vertices I and J, there is an arc from I to J, denoted by I −→ J, whenever I + K = J, for some ideal K in S. Also, the Cayley sum graph Cay + (I(R), S) is an undirected graph whose vertex set is the set I(R) and two distinct vertices I and J are adjacent whenever I + K = J or J + K = I, for some ideal K in S. In this paper, we study some basic properties of the graphs − − → Cay + (I(R), S) and Cay + (I(R), S) such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of Cay + (I(R), S) and also we provide some characterization for rings R whose Cayley sum graphs have genus one.
Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ′(R) is a graph... more Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ′(R) is a graph with vertex set W ∗(R), where W ∗(R) is the set of all nonzero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a / ∈ (b) and b / ∈ (a). In this paper, we study the primitive cycle of cozero-divisor graphs and we determine all Artinian rings with claw-free or triangle-free cozero-divisor graphs. Also, we investigate all Artinian rings whose cozero-divisor graphs are C4-free.
Let R be a commutative ring, I(R) be the set of all ideals of R and S be a subset of I * (R) = I(... more Let R be a commutative ring, I(R) be the set of all ideals of R and S be a subset of I * (R) = I(R) \ {0}. We define a Cayley sum digraph of ideals of R, denoted by − − → Cay + (I(R), S), as a directed graph whose vertex set is the set I(R) and, for every two distinct vertices I and J, there is an arc from I to J, denoted by I −→ J, whenever I + K = J, for some ideal K in S. Also, the Cayley sum graph Cay + (I(R), S) is an undirected graph whose vertex set is the set I(R) and two distinct vertices I and J are adjacent whenever I + K = J or J + K = I, for some ideal K in S. In this paper, we study some basic properties of the graphs − − → Cay + (I(R), S) and Cay + (I(R), S) such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of Cay + (I(R), S) and also we provide some characterization for rings R whose Cayley sum graphs have genus one.
Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ′(R) is a graph... more Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ′(R) is a graph with vertex set W ∗(R), where W ∗(R) is the set of all nonzero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a / ∈ (b) and b / ∈ (a). In this paper, we study the primitive cycle of cozero-divisor graphs and we determine all Artinian rings with claw-free or triangle-free cozero-divisor graphs. Also, we investigate all Artinian rings whose cozero-divisor graphs are C4-free.
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