Sharp bounds for sums associated to graphs of matrices

We provide a simple algorithm for finding the optimal upper bound for sums of products of matrix entries of the form Sπ(N) := N j1,...,j2m=1 ker j π t (1) j1j2 t (2) j3j4 ···t (m) j2m−1j2m where some of the summation indices are constrained to be equal. The upper bound is easily obtained from a graph Gπ associated to the constraints, π, in the sum. © 2011 Elsevier Inc. All rights reserved.