PQ-Type Adjacency Polytopes of Join Graphs

Abstract

PQ-type adjacency polytopes $$\nabla ^PQ _G$$ ∇ P Q G are lattice polytopes arising from finite graphs G. There is a connection between $$\nabla ^PQ _G$$ ∇ P Q G 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 $$\nabla ^PQ _G$$ ∇ P Q G plays a central role. In the present paper, we focus on the case where G is a join graph. In particular, formulas of the $$h^*$$ h ∗ -polynomial and the normalized volume of $$\nabla ^PQ _G$$ ∇ P Q G of a join graph G are presented. Moreover, we give explicit formulas of the $$h^*$$ h ∗ -polynomial and the normalized volume of $$\nabla ^PQ _G$$ ∇ P Q G when G is a complete multipartite graph or a wheel graph.

Laplacian polytopes of simplicial complexes

“…Over decades, several lattice polytopes arising from graphs have been studied, extensively. Prominent examples include matching polytopes, cut polytopes, edge polytopes, adjacency polytopes of several types, among which are symmetric edge polytopes (see e.g., [23,4,19,21,28,10]). Following this line of research, in 2017, Braun and Meyer [6] initiated the study of Laplacian simplices that are defined as the convex hull of the columns of the classical Laplacian matrix of a simple graph (see also [24,3]).…”