A perfect distance tree is a weighted tree with n vertices in which the set of distances between vertices is . We define a tree with n vertices to be a Perfect Distance Tree mod m if the distances (mod m) can be obtained. In this paper, we find that every star with m+1 vertices, where m is odd, labeled with 0, 1, 2, 3, ..., m-1 is a perfect distance tree mod m. The stars obtained from this star by removing the edge labeled 0 or by changing the weight 0 to another weight are also perfect distance trees mod m. By combining stars, we show that every star with km+j vertices can be labeled to be a perfect distance tree mod m, where m is odd, k ≥ 1 and -1 ≤ j ≤ 4. Finally, we show that certain twin-stars (trees of diameter 3) can be labeled as perfect distance trees mod m.

Author Bio

Xianrui Meng is double-majoring in Computer Science and Mathematics at Bloomsburg University of Pennsylvania. He is currently finishing his undergraduate work and will attend the PhD program in computer science at Boston University. Xianrui is from China and transferred to Bloomsburg University at the second year of his college. His hobbies include basket ball, music, travel, etc.