Combinatorics on words is an important branch of mathematics that has attracted considerable attention since the early 1900s, beginning with the work of Thue. In this talk, I will demonstrate how problems in this field can be solved using graph and group theory. The talk is divided into three parts: first, I will discuss pattern-counting in finite words using graph representations; second, I will address the uniform distribution property of automatic sequences; and finally, if time permits, I will present my approach to several conjectures concerning the Oldenburger-Kolakoski sequence.Polynomial eigenvalue problems arise in diverse areas of science and engineering, including structural vibration analysis, computer-aided geometric design, robotics, and machine learning. Their solution often relies on constructing suitable matrix linearizations that allow the use of established algorithms for the generalized eigenvalue problem.
