1 Nonsingularity of Divisor Tournaments

By Rohan Hemasinha Dept. of Math/Stat, Univ. of West Florida, Pensacola, FL 32514, USA rhemasin@uwf.edu Jeffrey L. Stuart Dept. of Mathematics, Pacific Lutheran Univ. Tacoma, WA 98447, USA jeffrey.stuart@plu.edu James R. Weaver (speaker) Dept. of Math/Stat, Univ. of West Florida, Pensacola, FL 32514, USA jweaver@uwf.edu.

Matrix theoretic properties and examples of divisor tournaments are discussed. Emphasis is placed on results and conjectures about the nonsingularity of the adjacency matrix for a divisor tournament. Definition: For an integer n > 2, the divisor tournament D(T_{n}) ( a directed graph on the vertices 2, 3, …, n) is defined by: i is adjacent to j if i divides j, otherwise j is adjacent to i for 2i < jn. No vertex is adjacent to itself. Definition: The adjacency matrix T_{n} of the directed graph D(T_{n}) with vertex set {2, 3, …, n} is the (n − 1)(n − 1) matrix [t_{ij}] defined by t_{ij} = 1 and t_{ji} = 0 if iOj, t_{ij} = 0 and t_{ji} = 1 if i&#8740; j for 2i < jn. t_{ii} = 0 for i ∈ {2, 3, …, n}.