Spectral radii of tournament matrices whose graphs are related by an arc reversal

Abstract

Given an irreducible tournament matrix t and a pair of distinct indices 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 condition 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 spectral 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 Toeplitz tournament matrices, and the resulting comparison sheds light on some conjectures of Brualdi and Li. Further, if T yields equality in a certain lower bound on the spectral radius of a tournament matrix, then for any i and j, we provide simple necessary and sufficient conditions for the spectral radius of T( i, j) to be larger than that of T, to be smaller than that of T, and to be equal to that of T.