Extremal numbers of positive entries of imprimitive nonnegative matrices (Q884416)

The paper deals with the maximun and minimun numbers of positive entries of imprimitive nonnegative matrices. A square nonnegative matrix \(A\) is said to be primitive if \(A^p\) is a positive matrix for some positive integer \(p\); otherwise \(A\) is called imprimitive. The imprimitivity index of \(A\), \(ind(A)\), is the number of eigenvalues of \(A\) whose moduli are equal to the spectral radius of \(A\). The author denotes by \(\sigma(A)\) the number of positive entries of a nonnegative matrix \(A\) and, for a real number \(x\), \([x]\) is the largest integer not exceeding \(x\). In this paper the author determines the maximum and minimum numbers of positive entries of an irreducible nonnegative matrix with a given imprimitivity index. Specifically, if \(\Gamma(n,k)\) denotes the set of \(n \times n\) irreducible nonnegative matrices with imprimitivity index \(k\), the author proves: \[ \max\{\sigma(A) : A \in \Gamma(n,k)\}= \begin{cases} [n^2/k] & \text{if }1\leq k \leq 4, \\ 2n-k+[(n-k)^2/4] &\text{if }k \geq 5 \end{cases} \] and \[ \min\{\sigma(A) : A \in \Gamma(n,k)\}= \begin{cases} n+1 \text{if }k < n, \\ n &\text{if }k=n. \end{cases} \] Finally, the author obtains an estimate of \(ind(A)\) in terms of \(\sigma(A)\).