Computing Volumes of Adjacency Polytopes via Draconian Sequences

TL;DR

This paper introduces a combinatorial approach to compute the volume of adjacency polytopes related to power networks, providing recurrences and formulas for specific graph classes, with conjectures for broader applicability.

Contribution

It rephrases volume computation as counting draconian sequences and derives recurrences and formulas for various graph classes, advancing understanding of adjacency polytope volumes.

Findings

Recurrences for networks with connectivity at most 1

Explicit formulas for outerplanar graphs

Conjecture for all outerplanar graphs

Abstract

Adjacency polytopes appear naturally in the study of nonlinear emergent phenomena in complex networks. The "PQ-type" adjacency polytope, denoted $\nabla^{\mathrm{PQ}}_G$ and which is the focus of this work, encodes rich combinatorial information about power-flow solutions in sparse power networks that are studied in electric engineering. Of particular importance is the normalized volume of such an adjacency polytope, which provides an upper bound on the number of distinct power-flow solutions. In this article we show that the problem of computing normalized volumes for $\nabla^{\mathrm{PQ}}_G$ can be rephrased as counting $D(G)$-draconian sequences where $D(G)$ is a certain bipartite graph associated to the network. We prove recurrences for all networks with connectivity at most $1$ and, for $2$-connected graphs under certain restrictions, we give recurrences for subdividing an edge and taking the join of an edge with a new vertex. Together, these recurrences imply a simple, non-recursive formula for the normalized volume of $\nabla^{\mathrm{PQ}}_G$ when $G$ is part of a large class of outerplanar graphs; we conjecture that the formula holds for all outerplanar graphs. Explicit formulas for several other (non-outerplanar) classes are given. Further, we identify several important classes of graphs $G$ which are planar but not outerplanar that are worth additional study.