ON SIGNLESS LAPLACIAN SPECTRUM OF THE ZERO DIVISOR GRAPHS OF THE RING Zn

For a finite commutative ring R with identity 1 not equal 0, the zero divisor graph Gamma(R) is a simple connected graph having vertex set as the set of nonzero zero divisors of R, where two vertices x and y are adjacent if and only if xy = 0. We find the signless Laplacian spectrum of the zero divisor graphs Gamma(Z(n)) for various values of n. Also, we find signless Laplacian spectrum of Gamma(Z(n)) for n = p(z) , z >= 2, in terms of signless Laplacian spectrum of its components and zeros of the characteristic polynomial of an auxiliary matrix. Further, we characterise n for which zero divisor graph Gamma(Z(n)) are signless Laplacian integral.