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Let R be a commutative ring with A ⁢ (R) its set of annihilating-ideals. The extended annihilating-ideal graph of R, denoted by AG ¯ ⁢ (R) , is an undirected graph with vertex set A ⁢ (R) * = A ⁢ (R) ∖ { 0 } and two vertices I 1 and I 2 are adjacent if and only if I 1 m ⁢ I 2 n = 0 with I 1 m ≠ 0 and I 2 n ≠ 0 , for some positive integers m and n. In this paper, we first study some basic properties of AG ¯ ⁢ (R) and then we investigate the relationship between the extended annihilating-ideal graph AG ¯ ⁢ (R) , the annihilator-ideal graph A I ⁢ (R) and the annihilating-ideal graph AG ⁢ (R) of a commutative ring R. [ABSTRACT FROM AUTHOR]

For a finite commutative ring R with identity 1 ≠ 0 , the weakly zero-divisor graph of R denoted as W Γ (R) is a simple undirected graph having vertex set as a set of non-zero zero-divisors of R and two distinct vertices a and b are adjacent if and only if there exist elements r ∈ ann (a) and s ∈ ann (b) satisfying the condition r s = 0 . The zero-divisor graph of a ring is a spanning sub-graph of the weakly zero-divisor graph. This article finds the normalized Laplacian spectra of the weakly zero-divisor graph W Γ (R) . Specifically, the investigation is carried out on the weakly zero-divisor graph W Γ (Z n) for various values of n. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, and  Z (R)  be the set of zero-divisors of R. The weakly zero-divisor graph of R denoted by  W Γ (R)  is an undirected (simple) graph with vertex set  Z (R) * , and two distinct vertices x and y are adjacent, if and only if there exist  r ∈ a n n (x)  and  s ∈ a n n (y) , such that  r s = 0 . Importantly, it is worth noting that  W Γ (R)  contains the zero-divisor graph  Γ (R)  as a subgraph. It is known that graph theory applications play crucial roles in different areas one of which is chemical graph theory that deals with the applications of graph theory to solve molecular problems. Analyzing Zagreb indices in chemical graph theory provides numerical descriptors for molecular structures, aiding in property prediction and drug design. These indices find applications in QSAR modeling and chemical informatics, contributing to efficient compound screening and optimization. They are essential tools for advancing pharmaceutical and material science research. This research article focuses on the basic properties of the weakly zero-divisor graph of the ring  Z p × Z t × Z s , denoted by  W Γ (Z p × Z t × Z s) , where p, t, and s are prime numbers that may not necessarily be distinct and greater than 2. Moreover, this study includes the examination of various indices and coindices such as the first and second Zagreb indices and coindices, as well as the first and second multiplicative Zagreb indices and coindices of  W Γ (Z p × Z t × Z s) . [ABSTRACT FROM AUTHOR]

Let R be a prime ring with char(R)≠2. A biadditive symmetric map Δ: R×R → R is called symmetric (α, β)-biderivation if, for any fixed y ∈ R, the map x ↦ Δ(x,y) is a (α, β)-derivation. A symmetric biadditive map Γ : R×R → R is a symmetric generalized (α, β)-biderivation if for any fixed y ∈ R, the map x ↦ Γ(x,y) is a generalized (α, β)-derivation of R associated with the (α, β)-derivation Δ(.,y). In the present paper, we investigate the commutativity of a ring having a generalized (α, β)-biderivation satisfying certain algebraic conditions. [ABSTRACT FROM AUTHOR]