For n a positive integer and v a vertex of a graphG, the nth order degree ofvinG, denoted bydegnv, is the number of vertices at distance n from v. The graphGis said to be nth order regular of degreekif, for every vertexvofG, degnv = k.The following conjecture due to Alavi, Lick, and Zou is proved: For n ≥ 2, ifGis a connected nth order regular graph of degree 1, thenGis either a path of length 2n—1 orGhas diameter n. Properties of nth order regular graphs of degreek, k ≥1, are investigated.