The Ehrhart and face polynomials of the graph polytope of a cycle

We are interested in the polytope consisting of all points in the first orthant such that the sum of two cyclically adjacent coordinates is less than or equal to 1. This polytope is also known as the graph polytope of a cycle. Using spectral techniques, we obtain a determinant for the Ehrhart quasi-polynomial of this polytope and hence also an expression for the volume of this polytope. The spectral techniques also yield a combinatorial expression for the face polynomial of this polytope in terms of matchings of a cycle.(c) 2023 Elsevier Ltd. All rights reserved.