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The scissors congruence conjecture for the unimodular group is an analogue of Hilberts third problem, for the equidecomposability of polytopes. Liu and Osserman studied the Ehrhart quasi-polynomials of polytopes naturally associated to graphs whose vertices have degree one or three. In this paper, we prove the scissors congruence conjecture, posed by Haase and McAllister, for this class of polytopes. The key ingredient in the proofs is the nearest neighbor interchange on graphs and a naturally arising piecewise unimodular transformation.
A graph whose nodes have degree 1 or 3 is called a ${1,3}$-graph. Liu and Osserman associated a polytope to each ${1,3}$-graph and studied the Ehrhart quasi-polynomials of these polytopes. They showed that the vertices of these polytopes have coordin
Let $P(b)subset R^d$ be a semi-rational parametric polytope, where $b=(b_j)in R^N$ is a real multi-parameter. We study intermediate sums of polynomial functions $h(x)$ on $P(b)$, $$ S^L (P(b),h)=sum_{y}int_{P(b)cap (y+L)} h(x) mathrm dx, $$ where w
The aim of the article is to understand the combinatorics of snake graphs by means of linear algebra. In particular, we apply Kasteleyns and Temperley--Fishers ideas about spectral properties of weighted adjacency matrices of planar bipartite graphs
This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree coefficients o
It was observed by Bump et al. that Ehrhart polynomials in a special family exhibit properties similar to the Riemann {zeta} function. The construction was generalized by Matsui et al. to a larger family of reflexive polytopes coming from graphs. We