Abstract A graph G is said to be retarded regular if there is a positive integral number s such that the number of walks of length s starting at vertices of G is a constant function. Regular and semiregular graphs are retarded regular with s = 1 and s\!≤ \!2, respectively. We prove that any retarded regular connected graph is either regular or semiregular. Keywords: Retarded regular graphsAdjacency matrixSemiregular graphAMS Subject Classifications:: 05C5005C75