Given integers c, k and d, minimum c-connected k-regular graphs with diameter d are considered. Minimum c-connected k-regular graphs with diameter d are the graphs with the fewest number of vertices among the c-connected k-regular graphs with diameter d.
For all integers c, k and d, we give the lower bound for the number of vertices of minimum c-connected k-regular graphs with diameter d. And for all integers c, k and d except i) d = 2, k ≥ 3, ii) d ≥ 3, k ≥ 3, k = 3c - 1 with odd c and iii) d = 5 + 3m (m ≥ 0), k ≥ 3, k ≥ 3c - 1 with even k and odd c, we present the construction method for a class of minimum c-connected k-regular graphs with diameter d in which the number of vertices is equal to the above lower bound.