On the coefficients of Coxeter polynomials of trees and bipartite quivers

We apply spectral graph theory and a theorem of A’Campo to express the first and second coefficients of the Coxeter polynomials associated with certain bipartite quivers in terms of the degrees of the vertices in their underlying graphs. As a consequence, we provide a new proof of a result by Happel, expressing the second coefficient of the Coxeter polynomial of a tree in terms of its vertex degrees.