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

TL;DR

This paper extends the understanding of normalized volumes of type-PQ adjacency polytopes for various graphs, providing new formulas and generalizing key recurrence relations to broader classes of graphs.

Contribution

It generalizes the triangle recurrence for normalized volumes and derives formulas for graphs formed by deleting paths or cycles from complete graphs.

Findings

Triangle recurrence applies in a more general setting.

Formulas for graphs with deleted paths or cycles.

Normalized volumes can be computed via integer sequences.

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.