A Homomorphic Polynomial for Oriented Graphs

In this article, we define a function that counts the number of (onto) homomorphisms of an oriented graph. We show that this function is always a polynomial and establish it as an extension of the notion of chromatic polynomials. We study algebraic properties of this function. In particular we show that the coefficients of these polynomials have the alternating sign property and that the polynomials associated to the independent sets have relations with the Stirling numbers of the second kind.