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For a finite commutative ring $\mathbb{Z}_{n}$ with identity $1\neq 0$, the zero divisor graph $\Gamma(\mathbb{Z}_{n})$ 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 distance Laplacian spectrum of the zero divisor graphs $\Gamma(\mathbb{Z}_{n})$ for different values of $n$. Also, we obtain the distance Laplacian spectrum of $\Gamma(\mathbb{Z}_{n})$ for $n=p^z$, $z\geq 2$, in terms of the Laplacian spectrum. As a consequence, we determine those $n$ for which zero divisor graph $\Gamma(\mathbb{Z}_{n})$ is distance Laplacian integral.

Let be the set of all matrices, where and are the sums of entries in row and column , respectively. Sampling efficiently uniformly at random elements of is a problem with interesting applications in Combinatorics and Statistics. To calibrate the statistic for testing independence, Diaconis and Gangolli proposed a Markov chain on that samples uniformly at random contingency tables of fixed row and column sums. Although the scheme works well for practical purposes, no formal proof is available on its rate of convergence. By using a canonical path argument, we prove that this Markov chain is fast mixing and the mixing time is given by where is the maximal value in a cell.

Let Ω be the set of all m × n matrices, where r i and c j are the sums of entries in row i and column j, respectively. Sampling efficiently uniformly at random elements of Ω is a problem with interesting applications in Combinatorics and Statistics. To calibrate the statistic χ 2 for testing independence, Diaconis and Gangolli proposed a Markov chain on Ω that samples uniformly at random contingency tables of fixed row and column sums. Although the scheme works well for practical purposes, no formal proof is available on its rate of convergence. By using a canonical path argument, we prove that this Markov chain is fast mixing and the mixing time τ x ( ϵ ) is given by τ x ( ϵ ) ≤ 2 c m a x ( m n ) 4 ln ( c m a x ϵ − 1 ) ,where c m a x − 1 is the maximal value in a cell. Keywords: Sampling, Canonical path, Mixing time of Markov chains