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

Abstract

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 ) .