On normalized Laplacian spectrum of zero di-visor graphs of commutative ring Zn

For a finite commutative ring Z(n) with identity 1 not equal 0, the zero divisor graph Gamma(Z(n)) is a simple connected graph having vertex set as the set of non-zero zero divisors, where two vertices x and y are adjacent if and only if xy = 0. We find the normalized Laplacian spectrum of the zero divisor graphs Gamma (Z(n))for various values of n and characterize n for which Gamma (Z(n)) is normalized Laplacian integral. We also obtain bounds for the sum of graph invariant S-beta* (G)-the sum of the beta-th power of the non-zero normalized Laplacian eigenvalues of Gamma (Z(n)).