In this appendix to the authors' paper [arXiv:1006.3807], we give conditions
which characterize the Minkowski measurability of a certain class of
self-similar tilings and (self-similar sets). Under appropriate hypotheses,
self-similar tilings with simple generators (more precisely, monophase
generators) are shown to be Minkowski measurable if and only if the associated
scaling zeta function is of nonlattice type.

This paper studies the "energy space" $\mathcal{H}_{\mathcal{E}}$ (the
Hilbert space of functions of finite energy, aka the Dirichlet-finite
functions) on an infinite network (weighted connected graph), from the point of
view of the multiplication operators $M_f$ associated to functions $f$ on the
network. We show that the multiplication operators $M_f$ are not Hermitian
unless $f$ is constant, and compute the adjoint $M_f^\star$ in terms of a
reproducing kernel for $\mathcal{H}_{\mathcal{E}}$.

Motivated by potential theory on discrete spaces, we study a family of
unbounded Hermitian operators in Hilbert space which generalize the usual
graph-theoretic discrete Laplacian. These operators are discrete analogues of
the classical conformal Laplacians and Hamiltonians from statistical mechanics.
For an infinite discrete set $X$, we consider operators acting on Hilbert
spaces of functions on $X$, and their representations as infinite matrices; the
focus is on $\ell^2(X)$, and the energy space $\mathcal{H}_{\mathcal E}$.