Volumes of consecutively defined sets

Abstract

We study a variant of the graph polytopes of a path and of a cycle where we replace the inequality $x_{i} + x_{i+1} \leq 1$ with the two inequalities $(1-\alpha) \cdot x_{i} + \alpha \cdot x_{i+1} \leq \alpha$ for $0 \leq x_{i} \leq \alpha$ and $\alpha \cdot x_{i} + (1-\alpha) \cdot x_{i+1} \leq \alpha$ for $\alpha \leq x_{i} \leq 1$. Using a self-adjoint operator and its eigenvalues we obtain convergent series for their volumes. As a corollary we obtain that the volumes of the set associated to a path on $n$ vertices and the set associated to a cycle on $n+1$ vertices are related by a constant factor of $\alpha$.