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.
Dr. William Calhoun, Department of Mathematics, Computer Science and Statistics, Bloomsburg Universityof Pennsylvania firstname.lastname@example.org
"Perfect Distance Stars mod m,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 11
, Article 5.
Available at: https://scholar.rose-hulman.edu/rhumj/vol11/iss1/5