A note on Perron vectors for almost regular tournament matrices (Q1372956)

A tournament matrix \(T\) of even order \(n\) with half of the row sums equal to \(n/2-1\) and half equal to \(n/2\) is called almost regular. The entry \(w_j\) of the Perron vector \(w\) of \(T\) can be interpreted as the strength of player \(j\). It is shown that if the \(i\)th and \(j\)th row sums of \(T\) are \(n/2-1\) and \(n/2\), respectively, then \(w_i<w_j\).