PQ-type adjacency polytopes of join graphs

Abstract

PQ-type adjacency polytopes ${\nabla }_G^{\mathrm{PQ}}$ are lattice polytopes arising from finite graphs $G$. There is a connection between ${\nabla }_G^{\mathrm{PQ}}$ and the engineering problem known as power-flow study, which models the balance of electric power on a network of power generation. In particular, the normalized volume of ${\nabla }_G^{\mathrm{PQ}}$ plays a central role. In the present paper, we focus the case where $G$ is a join graph. In fact, formulas of the $h^{\ast }$-polynomial and the normalized volume of ${\nabla }_G^{\mathrm{PQ}}$ of a join graph $G$ are presented. Moreover, we give explicit formulas of the $h^{\ast }$-polynomial and the normalized volume of ${\nabla }_G^{\mathrm{PQ}}$ when $G$ is a complete multipartite graph or a wheel graph.

arXiv comment18 pages, 1 figure

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Cited by in corpus (1)

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