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 formSπ(N):=∑j1,…,j2m=1kerj⩾πNtj1j2(1)tj3j4(2)⋯tj2m−1j2m(m) where some of the summation indices are constrained to be equal. The upper bound is easily obtained from a graph GπGπ associated to the constraints, π, in the sum.