Computing Volumes of Adjacency Polytopes via Draconian Sequences

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.

PQ-Type Adjacency Polytopes of Join Graphs

“…On the other hand, Theorem 1.1 is not useful for computing the h * -polynomial and the normalized volume of ∇ PQ G when G is a wheel graph W n , that is, G is the join of a cycle C n and K 1 . We give explicit formulas for the h * -polynomial and the normalized volume of ∇ PQ W n and prove the conjecture [8,Conj. 4.4] on the normalized volume of ∇ PQ W n (Theorem 5.1) by computing the perfectly matchable set polynomial of D(C n ).…”

“…Unfortunately, Theorem 1.1 is not useful for computing the h * -polynomial of ∇ PQ W n . We will give an explicit formula for the h * -polynomial of ∇ PQ W n and prove the conjecture [8,Conj. 4.4] on the normalized volume of ∇ PQ W n by using Proposition 2.6 on…”

“…, n}. Define D(G) to be the bipartite graph on [n] ∪ [n] with edges {i, i} for each i ∈ [n] and {i, j} and {i, j} for each edge {i, j} in G. It then follows that ∇ PQ G is unimodularly equivalent to Q D(G) [8,Lem. 2.4].…”

“…Moreover, Davis and Chen [8] have studied the normalized volume of ∇ PQ G by using draconian sequences.…”