The subset of the complex plane that consists of all eigenvalues of all stochastic matrices of a fixed order was completely determined by Karpelevich (1951). The boundary of this region consists of so-called Karpelevich arcs. Johnson and Paparella (2017)  made several conjectures on properties of these arcs. In this paper, we prove two of their conjectures. Specifically, we prove that the Karpelevich arcs are regular differentiable curves and establish that some powers of certain Karpelevich arcs correspond to some other Karpelevich arcs.
Bibliographical noteFunding Information:
We are grateful to the reviewer for valuable comments and suggestions, which greatly improved this paper. B. Kim's research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2B5B01001864 ).
© 2020 Elsevier Inc.
- Farey pair
- Karpelevich arc
- Stochastic matrix
ASJC Scopus subject areas
- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics