Graph Energy of Ideal-Based Zero-Divisor Graphs.

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.