Disconnected Vertex Sets and Equidistant Code Pairs

Two disjoint subsets A and B of a vertex set V of a flnite graph G are called disconnected if there is no edge between A and B .I f V is the set of words of length n over an alphabetf1;:::;qgand if two words are adjacent whenever their Hamming distance is not equal to afl xed ‐2f 1;:::;ng, then a pair of disconnected sets becomes an equidistant code pair. For disconnected sets A and B we will give a bound forjAj¢j Bj in terms of the eigenvalues of a matrix associated with G. In case the complement of G is given by a relation of an association scheme the bound takes an easy form, which applied to the Hamming scheme leads to a bound for equidistant code pairs. The bound turns out to be sharp for some values of q, n and ‐ ,a nd for q!1for any flxed n and ‐. In addition, our bound reproves some old results of Ahlswede and others, such as the maximal value ofjAj¢jBjfor equidistant code pairs A ans B in the binary Hamming Scheme.