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

Abstract

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

Keywords

• normalized laplacian matrix, normalized laplacian spectrum, zero divisor graph