On the variance of Shannon products of graphs

We study the combinatorial problem of finding an arrangement of distinct integers into the d-dimensional N-cube so that the maximal variance of the numbers on each image-dimensional section is minimized. Our main tool is an inequality on the Laplacian of a Shannon product of graphs, which might be a subject of independent interest. We describe applications of the inequality to multiple description scalar quantizers (MDSQ), to get bounds on the bandwidth of products of graphs, and to balance edge-colorings of regular, d-uniform, d-partite hypergraphs.