On the variance of Shannon products of graphs

We study the combinatorial problem of finding an arrangement of distinct integers into the ddimensional N-cube so that the maximal variance of the numbers on each -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 getting bounds on the bandwidth of products of graphs, and to balanced edge-colorings of regular, d-uniform, d-partite hypergraphs.

• Type: Journal
• Year: 2008
• Where: DAM
• Authors: József Balogh, Clifford D. Smyth
• Keywords: Combinatorial Problem, DAM 2008, Distinct Integers, Maximal Variance