The spectral classes of unicyclic graphs Original Research Article

We study the spectral classes of a finite, connected graph without loops and with exactly one “essential” cycle. A spectral class consists of all those orientations of the graph that do not contain oriented cycles and yield the same Coxeter polynomial. We show that, if the essential cycle has m vertices, then there are exactly [m/2] distinct spectral classes; the corresponding spectral radii are distinct in case the graph is wild. Furthermore, we give an explicit combinatorial expression for the entries of the powers of the Coxeter matrix of a finite quiver without oriented cycles.