SPECTRAL RADII OF TOURNAMENT MATRICES WHOSE GRAPHS ARE RELATED BY AN ARE REVERSAL

Given an irreducible tournament matrix T and a pair of distinct indice s i and j, let T(i,j) be the matrix obtained from T by transposing its principal submatrix on rows and columns i and j. We establish one con dition on rows i and j of T under which the spectral radius of T(i,j) is no smaller than that of T, and another condition on the ith and jth entries of the left and right Perron vectors of T under which the spe ctral radius of T(i,j) must be strictly smaller than that of T. These conditions are used to compare the spectral radii of a class of Toepli tz tournament matrices, and the resulting comparison sheds light on so me conjectures of Brualdi and Li. Further, if T yields equality in a c ertain lower bound on the spectral radius of a tournament matrix, then for any i and j, we provide simple necessary and sufficient condition s for the spectral radius of T(i, j) to be larger than that of T, to b e smaller than that of T, and to be equal to that of T.