Bounds on separated pairs of subgraphs, eigenvalues and related polynomials

We give a bound on the sizes of two sets of vertices at a given minimum distance (a separated pair of subgraphs) in a graph in terms of polynomials and the spectrum of the graph. We find properties of the polynomial optimizing the bound. Explicit bounds on the number of vertices at maximal distance and distance two from a given vertex, and on the size of two equally large sets at maximal distance are given, and we find graphs for which the bounds are tight.Graphs;Eigenvalues;Polynomials;mathematics