Automorphisms group of generalized Hamming Graphs

A generalized Hamming graph is a graph obtained by a cartesian product of different Hamming graphs. We provide in this paper, under some conditions, a characterization of these graphs using the automorphism group. Then, to study the vertex transitivity of some quasi-amply regular graphs, we give in first the properties of the automorphism group of a prime amply-regular graphs; then we exploit, particulary the main properties of the cartesian product, the prime factor decomposition and the notion of relatively prime graphs. Some graphs of abelian transitive Automorphism group are finally considered.