Vectorization hierarchies of some graph quantifiers

We give a sufficient condition for the inexpressibility of the k-th extende d vectorization of a generalized quantifier Q in FO(Q(k)), the extension of first-order logic by all k-ary quantifiers. The condition is based on a mo del construction which, given two FO(Q(1))-equivalent models with certain a dditional structure, yields a pair of FO(Q(k))-equivalent models. We also c onsider some applications of this condition to quantifiers that correspond to graph properties, such as connectivity and planarity.