Criteria for vanishing of Eulerian polynomials on n × n matrices

AbstractEulerian polynomial identities on n × n matrices were introduced by Szigeti, Tuza and Révész. Here we prove that some Eulerian polynomials are contained in the T-ideal generated by polynomials corresponding to graphs with less vertices. As a by-product we obtain a generalization of a graph-theoretic result of Swan. Then we give a complete description of Eulerian graphs with two vertices such that the corresponding identity is satisfied by the n × n matrix ring over a unitary commutative ring of characteristic 0