Normalized Volumes of Type-PQ Adjacency Polytopes for Certain Classes of Graphs

Abstract

The type-PQ adjacency polytope associated to a simple graph is a $0/1$-polytope containing valuable information about an underlying power network. Chen and the first author have recently demonstrated that, when the underlying graph $G$ is connected, the normalized volumes of the adjacency polytopes can be computed by counting sequences of nonnegative integers satisfying restrictions determined by $G$. This article builds upon their work, namely by showing that one of their main results -- the so-called "triangle recurrence" -- applies in a more general setting. Formulas for the normalized volumes when $G$ is obtained by deleting a path or a cycle from a complete graph are also established.