Eigenvalues, eigenspaces and distances to subsets

In this note we show how to improve and generalize some calculations of diameters and distances in sufficiently symmetrical graphs, by taking all the eigenvalues of the adjacency matrix of the graph into account. We present some applications of these results to the problem of finding tight upper bounds on the covering radius of error-correcting codes, when the weight distribution of the code (or the dual code) is known.