On distance-irregular labelings of cycles and wheels

Distance-irregular labeling was introduced by Slamin, and in his paper he determined the distance-irregular labeling for cycles and wheels of length f0; 1; 2; 5g mod 8. In this paper, we complete his results for cycles and wheels in general and prove the conjecture regarding the distance irregularity strength on wheels. We also show the general relation of the distance irregularity strength between a distance-irregular graph G and the graph G + K₁. Finally, we determine the distance irregularity strength of m-book graphs B_m.