| This dissertation studies three problems, each problem emphasizing a different aspect of the relationship between combinatorics, graph theory, and matrix theory.; The first problem concerns the existence of an n by n row-orthogonal (0, 1, -1)-matrix whose first row has no zeros. It is shown that a necessary condition of such a matrix is that n is not of the form pk, 2 pk or 3p where p is an odd prime, and k is a positive integer. In addition it is shown that the existence of a row-orthogonal (0, 1, -1)-matrix of order n with full column is equivalent to the existence of a Hadamard matrix of order n.; The second problem concerns the combinatorial structure of commuting pairs of matrices. The qualitative product of two sign patterns A&d14; and B&d14; is defined, and A&d14; and B&d14; are said to combinatorially commute if the qualitative product of A&d14; and B&d14; , and A&d14; and B&d14; , are equal. Two graphs combinatorially commute if their adjacency matrices combinatorially commute; two graphs allow commuting if there exist real symmetric matrices corresponding to these two sign patterns which commute. Two graphs are like-graphs if they are the same, except for possibly their loops. Pairs of like-trees which combinatorially commute and like-trees which allow commuting are completely classified. In addition we show that two (not necessarily like) paths on n ≥ 14 vertices without loops combinatorially commute if and only if they allow commuting.; Finally, the third problem concerns a two player game on a tournament T in which the players alternate moves by selecting arcs of T, and at each stage, if possible, extend the path created by the previous moves; the last player to move wins. Families of tournaments for which the first and second player to move has a winning strategy are found; and a winning strategy is outlined. |
