In this paper we address the Adjacent Quadratic Assignment Problem (AQAP) which is a variant of the QAP where the cost coefficient matrix has a particular structure. Motivated by strong lower bounds obtained by applying Reformulation Linearization Technique (RLT) to the classical QAP, we propose two special RLT representations for the problem. The first is based on a ``flow'' formulation whose linear relaxation can be solved very efficiently for large instances while the second one has significantly more variables and constraints, but possesses some desirable properties relative to the constraint set. Our computational results indicate that the second model outperforms the classical level-2 RLT, as it obtains lower bounds close to the optimal solutions for all instances with less computational effort.