PQ-type adjacency polytopes of join graphs

Ohsugi, Hidefumi; Tsuchiya, Asato

doi:10.48550/arxiv.2103.15045

Cited by 2 publications

(1 citation statement)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The first author, together with Chen and others, have recently studied their combinatorial structures and triangulations with a view towards their use in homotopy continuation methods [3,4,7]. Their work has sparked a flurry of results [1,5,6,10,11] using various algebraic and combinatorial techniques, often under the name of symmetric edge polytopes, which were studied previously with a view towards Ehrhart-theoretic results [8,9].…”

Section: Introductionmentioning

confidence: 99%

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

Davis¹,

Jakovleski²,

Pan³

2022

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

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

Davis¹,

Jakovleski²,

Pan³

2022

Preprint

View full text Add to dashboard Cite

show abstract

Computing Volumes of Adjacency Polytopes via Draconian Sequences

Davis

Chen

2022

Electron. J. Combin.

View full text Add to dashboard Cite

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.

show abstract

PQ-type adjacency polytopes of join graphs

Cited by 2 publications

References 16 publications

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

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

Computing Volumes of Adjacency Polytopes via Draconian Sequences

Contact Info

Product

Resources

About