  Let ${\bf A} \in R^{n \times n}$ be a nonnegative irreducible square matrix
and let $r({\bf A})$ be its spectral radius and Perron-Frobenius eigenvalue.
Levinger asserted and several have proven that $r(t):=r((1{-}t) {\bf A} + t
{\bf A}^\top)$ increases over $t \in [0,1/2]$ and decreases over $t \in
[1/2,1]$. It has further been stated that $r(t)$ is concave over $t \in (0,1)$.
Here we show that the latter claim is false in general through a number of
counterexamples, but prove it is true for ${\bf A} \in R^{2\times 2}$, weighted
shift matrices (but not cyclic weighted shift matrices), tridiagonal Toeplitz
matrices, and the 3-parameter Toeplitz matrices from Fiedler, but not Toeplitz
matrices in general. A general characterization of the range of $t$, or the
class of matrices, for which the spectral radius is concave in Levinger's
homotopy remains an open problem.
