PQ-Type Adjacency Polytopes of Join Graphs

Abstract

AbstractPQ-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.