The Polynomial Identities Of Eulerian Graphs And Matrix Rings

In this paper, by using the Swan’s graph theoretic method in the proof of Amitsur-Levitzki theorem, a new class of polynomial identities on the ring of matrices over the commutative ring was studied.On the one hand, the Eulerian polynomial was constructed by using the simple Eulerian graph and the condition which made this kind of polynomial to be a polynomial identity on Mn(C) was discussed on the basis of the the Swan’s graph theoretic method. Meanwhile, the minimum Eulerian identity was obtained and the conclusion of Chang and Giambruno-Sehgal was extended. On the other hand, according to the similar method that the identity was constructed by the Euler theory and by means of the calculations results of Aardenne-Ehenfest-Bruijn on Euler path, the condition which made the general symmetric polynomials to be a polynomial identity on Mn(C) was studied. Meanwhile, the minimum symmetric identity on Mn(C) was obtained and the conclusion of the Chang was extended.