A celebrated result of Karpelevič describes Θn, the collection of all eigenvalues arising from the stochastic matrices of order n. The boundary of Θn 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×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.