Grafos eulerianos e identidades polinomiais sobre a álgebra das matrizes

Abstract

In this work, we present applications of Graph Theory to the study of polynomial identities for the matrix algebra $M_n(K)$. Initially, we establish the fundamental concepts of PI-theory and Graph Theory, emphasizing the definition of Eulerian graphs. We analyze the Amitsur-Levitzki Theorem in detail through the proof proposed by Richard Swan [11], which utilizes the counting of Eulerian paths in directed graphs to simplify the algebraic problem. Subsequently, based on the work of Szigeti, Tuza, and Révész [13], we explore how specific graph constructions allow for the formalization of the conditions for the existence of these identities. Finally, we highlight that understanding this technique not only simplifies classical results but also establishes itself as a powerful tool for discovering new identities in PI-algebras.