A graph theoretical equivalence for Amitsur–Levitzki theorem

Abstract

Let n be a positive integer. A remarkable theorem of Amitsur–Levitzki in PI algebras states that if are any matrices with entries in any commutative ring, then , where is the standard polynomial over 2n variables . Now, let n and m be positive integers and let G be a multigraph with n vertices and m edges. Also consider a fixed labelling of edges, then any Eulerian trail in G (if exists), as a successive label of edges, corresponds to an element of with a specified parity sign. So, we may call an Eulerian trail as an odd or even Eulerian trail. R.G. Swan proved that when , then the number of even and odd Eulerian trails between any two vertices G (may be identical) are equal. We show that this latter theorem of Swan is a graph theoretical equivalence for Amitsur–Levitzki Theorem. Meanwhile, we present the Euler and Hamilton matrix of a graph with the entries denoting, respectively the number of possible Eulerian trails and Hamiltonian cycles of the graph.