Quasi-amply-regularity and Generalized Hamming Graphs

Let G λ be the class of the amply-regular graphs with parameters ( n , k , λ , 2 ) and G λ 1 , λ 2 be the one of the quasi-amply-regular graphs with parameters ( n , k , λ 1 , λ 2 , 2 ) . We first give some particular properties of G1 with new proofs and a new characterization of the Hamming graphs H(d, 3). Then under some conditions, we provides a characterization of the generalized Hamming graphs H ( d 1 , λ 1 + 2 ; d 2 , λ 2 + 2 ) obtained by a cartesian product of H ( d 1 , λ 1 + 2 ) and H ( d 2 , λ 2 + 2 ) as a quasi-amply-regular graphs with parameters ( n , k , λ 1 , λ 2 , 2 ) of maximum order for a given degree.