  A celebrated result of Karpelevi\v c describes $\Theta_n,$ the collection of
all eigenvalues arising from the stochastic matrices of order $n.$ The boundary
of $\Theta_n$ consists of roots of certain one-parameter families of
polynomials, and those polynomials are naturally associated with the so--called
reduced Ito polynomials of Types 0, I, II and III.
  In this paper we explicitly characterise all $n \times n$ stochastic matrices
whose characteristic polynomials are of Type 0 or Type I, and all sparsest
stochastic matrices of order $n$ whose characteristic polynomials are of Type
II or Type III. The results provide insights into the structure of stochastic
matrices having extreme eigenvalues.
