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).