2026· International Journal of Mathematical and Computer Sciences· 0 citations· 7 references
Abstract
Let R be a finite commutative ring with identity and let I be a proper ideal of R. The ideal-based zero-divisor graph \Gamma_I(R) has vertices outside I that annihilate some element outside I modulo I, with x adjacent to y whenever xy\in I. This paper studies the adjacency energy of \Gamma_I(R). General trace bounds are recorded, and an equitable-partition reduction is proved for ideal-filtered graphs with complete or null layers. For \Gamma_{(p^s)}({\mathbb Z}_{p^k}), valuation layers reduce the energy computation to the eigenvalues of an explicit threshold quotient matrix similar to a symmetric matrix. Closed formulas are obtained for s=2 and s=3, while the general prime-power case is reduced to a matrix of order s-1.
The zero divisor graph $\Gamma(R)$ of a finite commutative ring $R$ has as vertices the non-zero zero divisors of $R$, with an edge between two elements exactly when their product is zero. We determine the boxicity and threshold dimension of $\Gamma(R)$ for two classes of finite commutative rings: reduced rings and quo...
Let R be a commutative ring with the nonzero identity. The prime ideal sum graph of R, denoted by PIS(R), is a graph whose vertex-set is the set of all nonzero proper ideals of R, and two distinct vertices I and J are adjacent if and only if I + J is a prime ideal of R. In this study, our aim is to find out the topol...
Let R be a finite commutative ring with identity, and let Γ′ᵣ(R) denote its reduced cozero-divisor graph. We organize Γ′ᵣ(R) through the poset Prin*(R) of nonzero proper principal ideals and prove that adjacency is exactly incomparability in this poset. Consequently, cliques correspond to antichains, independent sets c...
Dong-Ze Du· Theoretical and Natural Scie...· 0 citations
This article examines the homological invariants, including Castelnuovo-Mumford regularity, projective dimension, and Betti numbers, of the edge ideals associated with the power graphs of integer modulo groups. We characterize the edge ideals of power graphs of group
\mathbb{Z}_{n}
with 2-linear resolution and list...
B. Rather, Jian-Feng Wang· Filomat· 2 citations· ⚡1
The zero-divisor graph of a commutative ring provides a natural connection between algebraic and graph-theoretic structures. Although extensive research has been conducted on the algebraic and combinatorial properties of zero-divisor graphs, their connectivity and metric properties over finite semilocal rings remain co...
Presley Kiplagat· Earthline Journal of Mathema...· 0 citations
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