  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 coordinates in the set $\{0,\frac14,\frac12,1\}$, which implies
that the period of their Ehrhart quasi-polynomials is either 1, 2, or 4. We
show that the period of the Ehrhart quasi-polynomial of these polytopes is at
most 2 if the graph is a tree or a cubic graph, and it is equal to 4 otherwise.
  In the process of proving this theorem, several interesting combinatorial and
geometric properties of these polytopes were uncovered, arising from the
structure of their associated graphs. The tools developed here may find other
applications in the study of Ehrhart quasi-polynomials and enumeration problems
for other polytopes that arise from graphs. Additionally, we have identified
some interesting connections with triangulations of 3-manifolds.
