On normalized Laplacian spectrum of zero divisor graphs of commutative ring ℤn

For a finite commutative ring  ℤ n  with identity  1 ≠ 0 , the zero divisor graph  Γ (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  Γ (ℤ n )  for various values of n and characterize n for which  Γ (ℤ n )  is normalized Laplacian integral. We also obtain bounds for the sum of graph invariant  S β * ( G ) -the  sum of the  β -th power of the non-zero normalized Laplacian eigenvalues  of  Γ (ℤ n ) .