The conditional diameter of a connected graph is defined as follows: given a property of a pair of subgraphs of , the so-called conditional diameter or -diameter measures the maximum distance among subgraphs satisfying . That is,In this paper we consider the conditional diameter in which requires that for all , for all , and for some integers and , where denotes the degree of a vertex x of , denotes the minimum degree and the maximum degree of . The conditional diameter obtained is called -diameter. We obtain upper bounds on the -diameter by using the k-alternating polynomials on the mesh of eigenvalues of an associated weighted graph. The method provides also bounds for other parameters such as vertex separators.