Computation of Some Graph Energies of the Zero-Divisor Graph Associated with the Commutative Ring Zp2 [x]/(x²)

Let be the commutative ring = Zp2 [x]/ x² with identity and Z∗ be the set of all non-zero zero-divisors of. Then, Γ is said to be a zero-divisor graph if and only if a b = 0 where a, b V (Γ) = Z∗ and (a, b) E(Γ). Let λ₁, λ₂, . . . , λn be the eigenvalues of the adjacency matrix, and let µ₁, µ₂, . . . , µn be the eigenvalues of the Laplacian matrix of Γ(R). Then we discuss the energy E ( Γ(R)) = μn |λi| and m are the order and size of Γ(R).