The Zero-Divisor Graphs Which Are Uniquely Determined By Neighborhoods

Implements of Graph Blow-Up in Co-Maximal Ideal Graphs

“…It is not hard to verify that B n is also isomorphic to the zero-divisor graph of the ring (in fact, the multiplicative semigroup) 2 n . See also [5,6] for Boolean rings and Boolean graphs. We can find that this kind of graphs are highly symmetric.…”

The Structure of Finite Local Principal Ideal Rings

“…It holds in particular for every finite local ring R with characteristic p. For the ring R, we can construct a coefficient field of R starting from any generator of the multiplicative cyclic group R/ * . See also [4,Theorem 3.2].…”

Characterizations of Three Classes of Zero-Divisor Graphs

“…Rings having zero-divisor graphs such that all vertices have unique complements are classified in [7,Theorem 2.5]. In [12], any nonempty simple graph is called uniquely determined if all distinct vertices have distinct neighborhoods; that is, N(v) = N(w) if and only if v = w. A characterization of rings whose zero-divisor graphs are uniquely determined is provided in [12,Theorem 2.5]. In this paper, we continue the investigations of [1,7,12].…”

“…, a n ). Recall that a ring R is a Boolean ring if r 2 = r for all r ∈ R. In [12,Theorem 2.5], it was shown that the zerodivisor graph of any commutative ring R is uniquely determined if and only if either R is a Boolean ring, or the total quotient ring of R (that is, the ring T(R) = R R\Z(R) ) is local and x 2 = 0 for all x ∈ Z(R). In [7,Theorem 2.5], it was shown that any commutative ring R is a Boolean ring if and only if either R is isomorphic to one of the rings in the set {Z 2 , Z 2 ⊕ Z 2 }, or R has at least three nonzero zero-divisors and every vertex of Γ(R) has a unique complement.…”