In 1989, F. R. K. Chung gave a construction for certain directed h-regular graphs of small diameter. Her construction is based on finite fields, and the upper bound on the diameter of these graphs is derived from bounds for certain very short character sums. Here we present two similar constructions that are based on properties of discrete logarithms and exponential functions in residue rings modulo a prime power. Accordingly, we use bounds for certain sums with additive and multiplicative characters to estimate the diameter of our graphs. We also give a third construction that avoids the use of bounds for exponential sums.