0.2cm 0.03cm red

# Extremals of determinants for Laplacians
on discrete surfaces

Paul Hafemann  University of Oldenburg, Germany  paul.hafemann@uni-
oldenburg.de and  Boris Vertman  University of Oldenburg, Germany
<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

In this note we pursue a discrete analogue of a celebrated theorem by Osgood,
Phillips and Sarnak, which states that in a fixed conformal class of
Riemannian metrics of fixed volume on a closed Riemann surface, the zeta-
determinant of the corresponding Laplace-Beltrami operator is maximized on a
metric of constant Gaussian curvature. We study the corresponding question for
the discrete cotan Laplace operator on triangulated surfaces. We show for some
types of triangulations that the determinant (excluding zero eigenvalues) of
the discrete cotan-Laplacian is stationary at discrete metrics of constant
discrete Gaussian curvature. We present some numerical calculations, which
suggest that these stationary points are in fact minima, strangely opposite to
the smooth situation. We suggest an explanation of this discrepancy.

###### Contents

  1. 1 Introduction and statement of the main results
  2. 2 Preliminary elements of discrete geometry
  3. 3 Auxiliary results on matrices, their inverses and determinants
  4. 4 Properties of strongly symmetric graphs
  5. 5 Stationary point of determinants for complete graphs
  6. 6 Stationary point of determinants for strongly symmetric graphs
  7. 7 Numerical illustrations of the stationary points
  8. 8 Discussion and outlook

##  1\. Introduction and statement of the main results

In their celebrated theorem, Osgood, Phillips and Sarnak [OPS88] have employed
Ricci flow to prove that in a fixed conformal class and for fixed volume, the
zeta regularized determinant of the Laplace-Beltrami operator on a closed
Riemann surface is maximal when the metric is of constant Gaussian curvature.
Their result can be phrased as follows.

######  Theorem 1.1 (Osgood, Philips and Sarnak).

The zeta-determinant of the Laplace-Beltrami operator on a compact Riemann
surface of fixed volume attains its local maxima at metrics of constant
Gaussian curvature.

There are discrete counterparts to all of the objects appearing in this
statement: the Riemann surface becomes a triangulation endowed with a discrete
metric for which discrete Laplace operators can be defined. There are well-
studied concepts like discrete Gaussian curvature on a triangulation. Thus it
makes sense to ask if there is a discrete counterpart to the Osgood-Phillips-
Sarnak theorem. Our main observation is that two particular classes of
triangulations admit an analogy to the smooth case, namely triangulations
defined by complete and by symmetric graphs.

###  1.1. Extremals of determinants for complete graphs

Our first main result studies determinants on complete graphs. The full
statement is presented below in Theorem 5.1 and reads informally as follows.

######  Theorem 1.2.

Let $M=(V,E,T)$ be a triangulation of a smooth closed Riemannian surface,
consisting of vertices $V$, edges $E$ and triangle faces $T$. Consider a
discrete metric $l:E\to(o,\infty)$ and impose the following conditions:

  1. (1)

the graph is complete, i.e. all pairs of vertices are connected by an edge.

  2. (2)

$M$ has no boundary, i.e. each edge is bounded by exactly two triangles.

  3. (3)

the discrete metric $l$ induces angles $\pi/3$ in each triangle 111In
particular, the discrete Gaussian curvature of $(M,l)$ is constant at all
vertices..

Then the determinant (excluding zero eigenvalues) of the cotan-Laplace
operator $\Delta=\Delta_{\text{cot}}$ on $(M,l)$ is stationary with respect to
variational changes of $l$.

Examples that satisfy the conditions of the theorem include: a) a
triangulation of the sphere $\mathbb{S}^{2}$: the tetrahedron, inducing the
complete graph $K_{4}$; b) the $7$-point triangulation of the $2$-torus
$\mathbb{T}^{2}=\mathbb{S}^{1}\times\mathbb{S}^{1}$: the Császár polyhedron,
inducing the complete graph $K_{7}$. Both triangulations are illustrated in
Figure 1.

11114455223367 Figure 1. Left: the simplest triangulation of $\mathbb{S}^{2}$:
the tetrahedron, inducing the complete graph $K_{4}$. Right: the $7$-point
triangulation of the $2$-torus: the Császár polyhedron, inducing the complete
graph $K_{7}$.

The Császár polyhedron with uniform side lengths is illustrated in Figure 2,
where the colouring indicates which sides are glued together.

Figure 2. Császár polyhedron with uniform edge lengths.

###  1.2. Extremals of determinants for incomplete symmetric graphs

Our second main result studies determinants on incomplete, but strongly
symmetric graphs. The full statement is presented below in Theorem 6.1 and
reads informally as follows.

######  Theorem 1.3.

Let $M=(V,E,T)$ be a triangulation of a smooth closed Riemannian surface,
consisting of vertices $V$, edges $E$ and triangle faces $T$. Consider a
discrete metric $l:E\to(o,\infty)$ and impose the following conditions:

  1. (1)

the graph is strongly symmetric in the sense of Definition 4.1, or more
generally satisfies the conditions laid out in Remark 6.2.

  2. (2)

$M$ has no boundary, i.e. each edge is bounded by exactly two triangles.

  3. (3)

the discrete metric $l$ induces angles $\pi/3$ in each triangle

Then the determinant (excluding zero eigenvalues) of the cotan-Laplace
operator $\Delta=\Delta_{\text{cot}}$ on $(M,l)$ is stationary with respect to
variational changes of $l$.

Examples that satisfy the conditions of the theorem include a $6$-point
triangulation of the sphere $\mathbb{S}^{2}$, the octahedron, and the
$9$-point triangulation of the $2$-torus. Both triangulations are illustrated
in Figure 3.

$1$$2$$3$$1$$1$$2$$3$$1$$1$$4$$7$$4$$7$$5$$6$$8$$9$ Figure 3. Left: a
$6$-point triangulation of $\mathbb{S}^{2}$, the octahedron. Right: the
$9$-point triangulation of the $2$-torus.

The $9$-point triangulation of the torus with uniform side lengths is
illustrated in Figure 4, where the colouring as before indicates which sides
are glued together. Another example, where the conditions of Theorem 1.3
apply, is the icosahedron.

Figure 4. The $9$-point triangulation of $\mathbb{T}^{2}$ with uniform edge
lengths.

###  1.3. Extremals of determinants are local minima

Our final main observation consists of numerical illustrations which indicate
that the stationary triangulations found above may be (local) minima. This
does not fit into the spirit of Theorem 1.1, where the constant curvature
metrics are actually global maxima. This shows that some crucially different
phenomena may arise in the discrete setting versus the smooth case.

######  Possible explanation 1.4.

Although an analogy with the Osgood-Phillips-Sarnak theorem is desirable, the
authors see no mathematical reason why this can be hoped for, even if the
discretization is refined sufficiently. Indeed, [IzKh20] establishes a quite
general asymptotic behaviour of discrete determinants (though for a different
discrete Laplacian) under specific refinements of the discretization. The
constant term in the asymptotics is the logarithm of the zeta-regularized
determinant and varies with the metric according to the Osgood-Phillips-Sarnak
theorem. However, the logarithmic term in the expansion depends non-trivially
on the angles of the various triangles in the triangulation and does not
satisfy a variational formula in any obvious way.

_Acknowledgements:_ The authors thank Daniel Grieser for his invaluable
comments and a careful reading of the manuscript. Furthermore we want to
gratefully acknowledge Daniel Grieser’s observation of an equivalent
characterization of strongly symmetric graphs, which we wrote down in
Proposition 4.2.

##  2\. Preliminary elements of discrete geometry

In this section we shall recall some fundamental notions from discrete
geometry that we are using throughout this note. Consider a closed Riemann
surface and its triangulation $(V,E,T)$ as e.g. in [Lee11, Theorem 5.36],
where $V$ denotes the set of vertices, $E$ the set of edges and $T$ the set of
triangles. The triangulation has _no boundary_ , i.e. each edge appears in the
boundary of exactly two triangles.

######  Definition 2.1.

A _discrete metric_ on a triangulation $(V,E,T)$ is a collection of lengths
for each edge, i.e. a function ${l\colon E\to(0,\infty)}$ such that for each
triangle $(i,j,k)\in T$ with vertices $i,j,k\in V$ and edges
$(i,j),(i,k),(j,k)\in E$ the lengths $l_{vw}=l(v,w)$ satisfy the triangle
inequalities

$\displaystyle l_{ij}$ $\displaystyle<l_{jk}+l_{ki},$ $\displaystyle l_{jk}$
$\displaystyle<l_{ki}+l_{ij},$ $\displaystyle l_{ki}$
$\displaystyle<l_{ij}+l_{jk}.$

Lengths of the three sides in an Euclidean triangle determine the angles and
the area uniquely. This way, the length functional, i.e. the discrete metric,
defines angles and area of the individual triangles of the triangulation and
hence paves the way for the next definition.

######  Definition 2.2.

Consider a triangulation $(V,E,T)$. Let $T_{i}\subset T$ denote the set of
triangles with $i\in V$ as a vertex. Denote by $\alpha_{i}(t)$ the angle in
$t\in T_{i}$ at that vertex. Define the _discrete Gaussian curvature_ $K_{i}$
at the vertex $i$ as the angle defect

$K_{i}=\pi-\sum_{t\in T_{i}}\alpha_{i}(t).$

In virtue of a discrete metric and its induced angles, we can furthermore
define a discrete Laplace operator whose weights are defined in terms of these
angles.

######  Definition 2.3.

Given a discrete metric on a triangulation without boundary, the discrete
_cotan-Laplace operator_ $\Delta_{\text{cot}}$ is a linear operator acting on
functions $f:V\to\mathbb{R}$ as follows. For any vertex $i\in V$ and emanating
edge $(i,j)\in E$ consider the two angles $\alpha_{ij}$ and $\beta_{ij}$
opposite to the edge $(i,j)$ as in Figure 5. Let $A_{i}$ denote the so-called
”local area element”, given by one third of the total area of all triangles
with a vertex at $i\in V$. We define the edge weights

$w_{ij}=\cot\alpha_{ij}+\cot\beta_{ij}.$

  1. (1)

Then the cotan Laplacian is defined by

$(\Delta_{\text{cot}}f)(i):=\sum_{(i,j)\in E}w_{ij}(f(i)-f(j)).$

  2. (2)

The _”local area normalized”_ cotan Laplacian is defined by

$(\Delta^{\prime}_{\text{cot}}f)(i):=A_{i}^{-1}\sum_{(i,j)\in
E}w_{ij}(f(i)-f(j)).$

$l_{ij}$$\alpha_{ij}$$\beta_{ij}$$j$$i$ Figure 5. $\alpha_{ij}$ and
$\beta_{ij}$ in the formula for the cotan-Laplace operator.

One can enumerate the vertices of a graph, so that a function $f\colon
V\to{\mathbb{R}}$ is represented by a vector which we also call $f$. The
action of $\Delta_{\text{cot}}$ is then given by the matrix (also denoted by
$\Delta_{\text{cot}}$) with entries ( $w_{ij}:=0$ if $(i,j)\notin E$ )

$\displaystyle\begin{cases}&(\Delta_{\text{cot}})_{ii}=\sum\limits_{(i,j)\in
E}w_{ij},\qquad\quad(\Delta^{\prime}_{\text{cot}})_{ii}=\sum\limits_{(i,j)\in
E}w_{ij}A^{-1}_{i},\\\ &(\Delta_{\text{cot}})_{ij}=-w_{ij},\ \text{if $i\neq
j$},\quad(\Delta^{\prime}_{\text{cot}})_{ij}=-w_{ij}A^{-1}_{i},\ \text{if
$i\neq j$}.\end{cases}$ (2.1)

######  Definition 2.4.

We define the determinants $\det^{\prime}\Delta_{\text{cot}}$ and
$\det^{\prime}\Delta^{\prime}_{\text{cot}}$ as the product of the nonzero
eigenvalues of the respective matrices in (2.1). This is well-defined, since
the eigenvalues of (2.1) do not depend on the order in which the vertices are
enumerated.

The cotan-Laplacian is widely considered to be a ’good’ choice of a geometric
discrete Laplace operator that can approximate the classic Laplace-Beltrami
operator on a Riemann surface. Moreover, $\Delta_{\text{cot}}$ appears in the
evolution equation for the curvature $\\{K_{i}\\}$ along discrete Ricci flow,
as noted in [SSP08]. However, we remark that there are various other choices
with different weights $w_{ij}$.

##  3\. Auxiliary results on matrices, their inverses and determinants

###  3.1. An auxiliary result on the derivative of determinants

In order to calculate the derivative of the discrete determinant in the proofs
our two main results, Theorems 1.2 and 1.3 (corresponding to Theorems 5.1 and
6.1 below), we will use the following well-known formula.

######  Lemma 3.1.

For any differentiable family
$(L(t))_{t\in{\mathbb{R}}}\subset{\mathbb{R}}^{n\times n}$ of positive semi-
definite symmetric matrices of constant rank, the derivative of its
determinant is given by

$\frac{d}{dt}\log\operatorname{det^{\prime}}L(t)=\lim_{\delta\to
0}\operatorname{Tr}(\dot{L}(t)(L(t)+\delta)^{-1}).$

###### Proof.

For any differentiable one-parameter families of matrices
$B(t),C(t)\in{\mathbb{R}}^{n\times n}$ and $M(t):=B(t)C(t)$ it holds that

$\displaystyle\operatorname{Tr}(\dot{M}M^{-1})=\operatorname{Tr}((\dot{B}C+B\dot{C})C^{-1}B^{-1})=\operatorname{Tr}(\dot{B}B^{-1})+\operatorname{Tr}(\dot{C}C^{-1}),$
(3.1)

where e.g. $\dot{M}$ denotes differentiation of $M(t)$ in $t$. Thus, if
$B=B(t)$ is a family of invertible matrices and $C=B(t)^{-1}$, the following
expression vanishes (here $M(t):=B(t)B^{-1}(t)\equiv I$ with $I$ being the
identity matrix):

$\displaystyle\operatorname{Tr}(\dot{B}B^{-1})+\operatorname{Tr}\left(B\left(\frac{d}{dt}B^{-1}\right)\right)=\operatorname{Tr}(\dot{I}I^{-1})=0.$
(3.2)

For diagonalizable matrices $M(t)$, written as $M(t)=Q^{-1}(t)D(t)Q(t)$ for a
diagonal matrix family $D(t)$, we obtain using (3.1) twice and (3.2) once

$\displaystyle\operatorname{Tr}(\dot{M}(M^{-1}))=\operatorname{Tr}(\dot{D}D^{-1})+\operatorname{Tr}\left(\left(\frac{d}{dt}Q^{-1}\right)Q\right)+\operatorname{Tr}(\dot{Q}Q^{-1})=\operatorname{Tr}(\dot{D}D^{-1}).$
(3.3)

We now consider $M(t):=L(t)$ with its diagonalization

$D(t):=\textup{diag}(\lambda_{1}(t),\dots,\lambda_{n}(t)),$

with the diagonal of eigenvalues of $A(t)$. We shall enumerate
$\lambda_{m}(t),\dots,\lambda_{n}(t)$ as the positive eigenvalues with
$m-1=\dim\ker L(t)$ independent of $t$ by assumption. Note that from
$\lambda_{1}(t),\dots,\lambda_{m-1}(t)\equiv 0$ it follows that
$\dot{\lambda}_{1},\dots,\dot{\lambda}_{m-1}=0$. We compute using (3.3)

$\displaystyle\lim\limits_{\delta\to
0}\operatorname{Tr}(\dot{L}(L+\delta)^{-1})$
$\displaystyle=\lim\limits_{\delta\to
0}\operatorname{Tr}(\dot{D}(D+\delta)^{-1})=\lim\limits_{\delta\to
0}\sum_{i=0}^{n}\frac{\dot{\lambda}_{i}}{\lambda_{i}+\delta}$
$\displaystyle=\lim\limits_{\delta\to
0}\sum_{i=m}^{n}\frac{\dot{\lambda}_{i}}{\lambda_{i}+\delta}+\frac{0}{\delta}=\sum_{i=m}^{n}\frac{\dot{\lambda}_{i}}{\lambda_{i}}$
$\displaystyle=\sum_{i=m}^{n}\frac{d}{dt}\log(\lambda_{i})=\frac{d}{dt}\log\left(\prod_{i=m}^{n}\lambda_{i}\right)$
$\displaystyle=\frac{d}{dt}\log\operatorname{det^{\prime}}L.$

∎

###  3.2. An auxiliary result on traces of matrix products

In view of Lemma 3.1, we need auxiliary results that will allow us to evaluate
the trace of products of the form $\dot{L}(t)(L(t)+\delta)^{-1}$. The first
such auxiliary result deals with the trace of products of certain matrices.

######  Lemma 3.2.

Let $A,B\in{\mathbb{R}}^{n\times n}$ be symmetric matrices, such that

  1. (1)

$A=(a_{ij})$ is trace-free with $\sum_{i\neq j}a_{ij}=0$,

  2. (2)

$B=(b_{ij})$ satisfies $b_{ii}=x$, and $b_{ij}=y$, $i\neq j$, whenever
$a_{ij}\neq 0$.

Then $AB$ is also trace-free.

###### Proof.

The claim follows by a simple calculation

$\displaystyle\operatorname{Tr}(AB)$
$\displaystyle=\sum_{i}\sum_{j}a_{ij}b_{ji}=\sum_{i}\left(a_{ii}b_{ii}+\sum_{j,\>j\neq
i}a_{ij}b_{ji}\right)$ $\displaystyle=x\sum_{i}a_{ii}+y\sum_{i\neq
j}a_{ij}=0.$

∎

###  3.3. Explicit structure of some inverse matrices

The next result is needed to check if $AB:=\dot{L}(t)(L(t)+\delta)^{-1}$, with
$L(t)$ being the family of cotan-Laplacians, has the structure required in
Lemma 3.2 in case of complete graphs, as needed in Theorem 5.1.

######  Lemma 3.3.

Let $B=(b_{ij})\in{\mathbb{R}}^{n\times n}$ be a symmetric matrix with
$b_{ii}=x$, and $b_{ij}=y$, if $i\neq j$. Assume that $(x-y)(x+(n-1)y)\neq 0$.
Then $B^{-1}=(c_{ij})$ exists with

$c_{ii}=\frac{x+(n-2)y}{(x-y)(x+(n-1)y)},\ c_{ij}=-\frac{y}{(x-y)(x+(n-1)y)}\
\text{for}\ i\neq j,$

###### Proof.

Consider the matrix $C^{\prime}=(c^{\prime}_{ij})\in{\mathbb{R}}^{n\times n}$
with entries

$c^{\prime}_{ii}=x+(n-2)y,\ c^{\prime}_{ij}=-y\ \text{for}\ i\neq j,$

and compute

$\displaystyle(BC^{\prime})_{ii}$
$\displaystyle=\sum_{j}b_{ij}c^{\prime}_{ji}=b_{ii}c^{\prime}_{ii}+\sum_{j,\>j\neq
i}c^{\prime}_{ij}b_{ji}$
$\displaystyle=x(x+(n-2)y)-(n-1)y^{2}=(x-y)(x+(n-2)y),$
$\displaystyle(BC^{\prime})_{ij}$
$\displaystyle=\sum_{k}b_{ik}c^{\prime}_{kj}=b_{ii}c^{\prime}_{ij}+b_{ij}c^{\prime}_{jj}+\sum_{k\neq
i,j}b_{ik}c^{\prime}_{kj}$ $\displaystyle=-xy+y(x+(n-2)y)-(n-2)y^{2}=0\
\text{for}\ i\neq j.$

We conclude that the inverse $B^{-1}$ exists with

$B^{-1}=\frac{C^{\prime}}{(x-y)(x+(n-1)y)}.$

∎

##  4\. Properties of strongly symmetric graphs

The content of this section is needed to check if
$AB:=\dot{L}(t)(L(t)+\delta)^{-1}$, with $L(t)$ being the family of cotan-
Laplacians, has the structure required in Lemma 3.2 in case of graphs that are
not necessarily complete. Instead of completeness, we introduce some symmetry
conditions that will be assumed in Theorem 6.1 below. We shall begin with a
definition of such symmetry conditions here.

######  Definition 4.1.

Consider a finite graph222 This graph need not arise from a triangulation of a
smooth surface at the moment.  $(V,E)$ consisting of vertices $V$ and edges
$E\neq\varnothing$. We assume that the graph is connected, i.e. for any two
vertices $i,j\in V$ there exists a path of edges, connecting $i$ and $j$. We
say that two vertices are ”neighbors”, if they are connected by an edge. We
introduce some notation.

  1. (1)

$L(i,j)$ is the number of edges in a shortest path from $i$ to $j$ (the
”distance”),

  2. (2)

$\alpha_{s}(i,j)$ is the number of neighbors $k$ of $i$ that have distance $s$
from $j$.

We call a graph _strongly symmetric_ if for any $s\in{\mathbb{N}}_{0}$

$L(i,j)=L(v,w)\Rightarrow\alpha_{s}(i,j)=\alpha_{s}(v,w).$

We then write $\alpha_{s}(\ell):=\alpha_{s}(i,j)$ for $\ell=L(i,j)$.

Note that $L(i,j)=0$ iff $i=j$ and $L(i,j)=1$ iff $(i,j)\in E$. For given
$i,j\in V$ with $L(i,j)=\ell$, the longest possible distance of a vertex $k\in
V$ with $(i,k)\in E$ to $j$ is $L(k,j)=\ell+1$, since one can always choose
the $1$-edge path from $k$ back to $i$ followed by a shortest path from $i$ to
$j$. On the other hand, the shortest possible distance from such $k$ to $j$ is
$L(k,j)=\ell-1$ because if $L(k,j)\leq\ell-2$, this would contradict
$L(i,j)=\ell$ by the same reasoning. Hence for any $\ell$

$\alpha_{s}(\ell)\neq 0\ \Rightarrow\ s\in\\{\ell-1,\ell,\ell+1\\}.$ (4.1)

Strong symmetry moreover implies that the number of edges, emanating from each
$i\in V$, i.e. the degree of a vertex, is the same for each $i$ and is given
for each $\ell$ by

$\alpha_{1}(i,i)\equiv\alpha_{1}(0)=\alpha_{\ell}(\ell)+\alpha_{\ell-1}(\ell)+\alpha_{\ell+1}(\ell).$
(4.2)

Finally, for a finite graph we may define $L:=\max\\{L(i,j)\mid i,j\in V\\}$
and find

$\alpha_{L+1}(L)=0,\quad\alpha_{L}(L)=\alpha_{1}(0)-\alpha_{L-1}(L).$ (4.3)

Examples of a strongly symmetric graph are the octahedron and the $9$-point
triangulation of the $2$-torus as both shown in Figure 3. It will be of
advantage to employ an equivalent characterization of strongly symmetric
graphs.

######  Proposition 4.2.

A finite connected graph $(V,E)$ is strongly symmetric if and only if the
number $p_{m}(i,j)$ of (obviously not necessarily shortest) paths of length
$m\in{\mathbb{N}}$ between vertices $i,j\in V$ depends only on the distance
$\ell=L(i,j)$ between $i$ and $j$. In such a case we write

$p_{m}(i,j)=p_{m}(\ell).$

###### Proof.

Assume that $(V,E)$ is strongly symmetric. Obviously, $p_{1}(i,j)=1$ if $i$
and $j$ are neighbors, i.e. connected by an edge, and zero otherwise. Hence
$p_{1}(i,j)$ in fact depends only on the distance $\ell=L(i,j)$. Assume by
induction that $p_{m-1}(i,j)$ depends only on the distance $\ell=L(i,j)$ for
some $m\in{\mathbb{N}}$. Then for any $i,j\in V$ with $\ell=L(i,j)$ we find

$\displaystyle p_{m}(i,j)$ $\displaystyle=\sum_{s}\alpha_{s}(\ell)p_{m-1}(s)$
$\displaystyle $
$\displaystyle=\alpha_{\ell-1}(\ell)p_{m-1}(\ell-1)+\alpha_{\ell}(\ell)p_{m-1}(\ell)+\alpha_{\ell+1}(\ell)p_{m-1}(\ell+1).$

By this formula, $p_{m}(i,j)$ also depends only on the distance $\ell=L(i,j)$.
Hence this is true for all $m$. Conversely, assume that $p_{m}(i,j)$ depends
only on the distance $\ell=L(i,j)$. We shall show that the graph is strongly
symmetric. For this we note

$\displaystyle p_{\ell}(\ell)=\alpha_{\ell-1}(i,j)p_{\ell-1}(\ell-1),$
$\displaystyle
p_{\ell+1}(\ell)=\alpha_{\ell-1}(i,j)p_{\ell}(\ell-1)+\alpha_{\ell}(i,j)p_{\ell}(\ell),$
$\displaystyle
p_{\ell+2}(\ell)=\alpha_{\ell-1}(i,j)p_{\ell+1}(\ell-1)+\alpha_{\ell}(i,j)p_{\ell+1}(\ell)+\alpha_{\ell+1}(i,j)p_{\ell+1}(\ell+1).$

From here we conclude that $\alpha_{\ell\pm 1}(i,j)$ and $\alpha_{\ell}(i,j)$
indeed depend only on $\ell=L(i,j)$ and hence the graph is strongly symmetric.
∎

Proposition 4.2 does not provide a feasible way to check if a given graph is
actually strongly symmetric, since in general we cannot study the number
$p_{m}(i,j)$ of paths for _arbitrary_ length $m\in{\mathbb{N}}$. However,
Proposition 4.2 allows for a simple proof of the following lemma, which is an
analogue of Lemma 3.3 and is key for the proof of Theorem 6.1.

######  Lemma 4.3.

Let $G=(V,E)$ be a strongly symmetric graph. Enumerate the vertices
$V=\\{1,\dots,n\\}$ and consider for any $x,y\in{\mathbb{C}}$ a matrix
$B=(b_{ij})\in{\mathbb{C}}^{n\times n}$ with

$\displaystyle\begin{cases}b_{ii}=x,\>&i\in V,\\\ b_{ij}=y,\>&(i,j)\in E,\\\
b_{ij}=0,\>&(i,j)\notin E,\,i\neq j.\end{cases}$

Then, for any given $y$, $B$ is invertible for almost all $x$ and the inverse
$B^{-1}=(c_{ij})$ satisfies: $c_{ij}$ depends only on $L(i,j)$ and $x,y$. In
particular, there exist real-valued functions $x_{0}(x,y)$ and $x_{1}(x,y)$
such that

$\forall i\in V:c_{ii}=x_{0}(x,y),\quad\forall(i,j)\in E:c_{ij}=x_{1}(x,y).$

###### Proof.

We can write the matrix $B=B(x,y)$ as $B(x,y)=xI+yA$, where $I$ is the
identity $n\times n$ matrix and $A=(a_{ij})$ with $a_{ij}=1$ if $(i,j)\in E$
and zero otherwise. Using the cofactor formula, $B(x,y)^{-1}$ is a meromorphic
function in $x$, for a fixed $y$. For $|x|\geq|y|$ we can write using Neumann
series

$B(x,y)^{-1}=x^{-1}\sum_{m=0}^{\infty}(-1)^{m}\left(\frac{y}{x}\right)^{m}A^{m}.$

Note that the individual entries of $A^{m}=(a^{m}_{ij})$ can be written as

$a^{m}_{ij}=\sum_{i_{1},...,i_{m-1}}a_{ii_{1}}a_{i_{1}i_{2}}\cdots
a_{i_{m-1}j}.$

Since $a_{ij}=1$ if $(i,j)\in E$ and zero otherwise, $a^{m}_{ij}=p_{m}(i,j)$
is simply the number of paths of length $m$ from $i$ to $j$, in the notation
of Proposition 4.2. Consequently the value of the individual entries of
$B(x,y)^{-1}=(c_{ij}(x,y))$ with $L(i,j)=\ell$ fixed, depends only on $(x,y)$
and $\ell$. In particular, given any vertex pairs $(i,j)$ and
$(i^{\prime},j^{\prime})$ with $L(i,j)=L(i^{\prime},j^{\prime})$, we conclude
that $c_{ij}(x,y)=c_{i^{\prime}j^{\prime}}(x,y)$ for $|x|\geq|y|$. Hence
$c_{ij}(\cdot,y)\equiv c_{i^{\prime}j^{\prime}}(\cdot,y)$ as meromorphic
functions. This proves the statement. ∎

We close the section with an alternative proof of the same statement, without
using Proposition 4.2.

###### Alternative proof of Lemma 4.3.

We construct an inverse of $B$ by making the ansatz that $B^{-1}=(c_{ij})$ has
entries $c_{ij}=x_{\ell}\equiv x_{\ell}(x,y)$, where $\ell=L(i,j)$.
Considering the diagonal terms, $BC=\operatorname{Id}$ yields the equation

$(BC)_{ii}=b_{ii}c_{ii}+\sum_{i\neq
k}b_{ik}c_{ki}=xx_{0}+\alpha_{1}(0)yx_{1}=1.$ (4.4)

For the non-diagonal terms, consider first $(i,j)\in E$. Then $L(i,j)=1$ and

$(BC)_{ij}=b_{ii}c_{ij}+b_{ij}c_{jj}+\sum_{k\neq
i,j}b_{ik}c_{kj}=xx_{1}+yx_{0}+\alpha_{1}(1)yx_{1}+\alpha_{2}(1)yx_{2}=0.$
(4.5)

For general $i,j\in V$ with $L(i,j)=\ell\in[2,L-1]$ we obtain using (4.1)

$\begin{split}(BC)_{ij}&=b_{ii}c_{ij}+b_{ij}c_{jj}+\sum_{k\neq
i,j}b_{ik}c_{kj}\\\
&=xx_{\ell}+\alpha_{\ell-1}(\ell)yx_{\ell-1}+\alpha_{\ell}(\ell)yx_{\ell}+\alpha_{\ell+1}(\ell)yx_{\ell+1}=0.\end{split}$
(4.6)

Let now $L(i,j)=L$, i.e. $i$ and $j$ are two vertices of maximal distance
between each other. In this case we obtain using (4.3)

$(BC)_{ij}=xx_{L}+\alpha_{L-1}(L)yx_{L-1}+\alpha_{L}(L)yx_{L}=0.$ (4.7)

We shall solve the system (4.4), (4.5), (4.6) and (4.7) recursively. Note that
the number $\alpha_{1}(0)$ of emanating edges from each vertex in a strongly
symmetric graph is positive, since we assume $E\neq\varnothing$. Moreover
$a_{\ell+1}(\ell)\neq 0$ for $\ell<L$. Indeed, given $i,j\in V$ with maximal
distance $L(i,j)=L$, consider the corresponding path of edges. Enumerate the
vertices along that path by $i=i_{0},\cdots,i_{L}=j$. Then
$L(i_{L-\ell},i_{L})=\ell$ and $L(i_{L-\ell-1},i_{L})=\ell+1$. Thus, by
definition, $a_{\ell+1}(\ell)\neq 0$ for $\ell<L$ and $a_{L+1}(L)=0$. We then
obtain (assume without loss of generality that $y\neq 0$, since for $y=0$ the
matix $B=xI$ and the statement is trivial)

$\displaystyle\mbox{\eqref{recur-1}}\, \Rightarrow\, $ $\displaystyle
x_{1}=\frac{1-xx_{0}}{\alpha_{1}(0)y},$ $\displaystyle\mbox{\eqref{recur-3}}\,
\Rightarrow\, $ $\displaystyle
x_{\ell+1}=-\frac{xx_{\ell}+\alpha_{\ell-1}(\ell)yx_{\ell-1}+\alpha_{\ell}(\ell)yx_{\ell}}{\alpha_{\ell+1}(\ell)y},\,2\leq\ell\leq
L-1,$ $\displaystyle\mbox{\eqref{recur-3}}\, \Rightarrow\, $ $\displaystyle
x_{L}=-\frac{\alpha_{L-1}(L)yx_{L-1}}{\alpha_{L}(L)y+x}.$ (4.8)

While the first two lines in (4.8) above can be used to express
$x_{1},\cdots,x_{L}$ in terms of $x_{0}$, the third line allows us to compute
$x_{0}$ in terms of $x$ and $y$, thereby giving an explicit solution to all
$x_{0},\cdots,x_{L}$. Let us be precise. The recursion, i.e. the second
equation in (4.8), yields (we shall consider $y\neq 0$ fixed and make the
dependence on $x$ explicit)

$x_{\ell}=p^{\prime}_{\ell}(x)x_{0}+p^{\prime\prime}_{\ell-1}(x),$

where $p^{\prime}_{\ell}(x)$ and $p^{\prime\prime}_{\ell}(x)$ are polynomials
in $x$ of degree $\ell$. The leading coefficient of $p^{\prime}_{\ell}(x)$ is
non-zero for $\ell\leq L-1$ and is given explicitly by

$\displaystyle(-1)^{\ell}y^{-\ell}\left(\prod\limits_{k=0}^{\ell}\alpha_{k+1}(k)\right)^{-1}$
(4.9)

Now, plugging this into the last equation in (4.8) yields

$\displaystyle
p^{\prime}_{L}(x)x_{0}+p^{\prime\prime}_{L-1}(x)=-\frac{\alpha_{L-1}(L)y}{\alpha_{L}(L)y+x}\Bigl{(}p^{\prime}_{L-1}(x)x_{0}+p^{\prime\prime}_{L-2}(x)\Bigr{)}$
$\displaystyle\Rightarrow
x_{0}=-\frac{p^{\prime\prime}_{L-1}(x)\bigl{(}\alpha_{L}(L)y+x\bigr{)}+\alpha_{L-1}(L)yp^{\prime\prime}_{L-2}(x)}{p^{\prime}_{L}(x)\bigl{(}\alpha_{L}(L)y+x\bigr{)}+\alpha_{L-1}(L)yp^{\prime}_{L-1}(x)}=:\frac{q_{L}(x)}{q_{L+1}(x)}.$

The crucial aspect to note is that the denominator in the formula for $x_{0}$
is a polynomial of degree $(L+1)$ with non-zero leading coefficient (4.9).
Hence the formula for $x_{0}$ makes sense, and the system of equations (4.8)
can be solved uniquely for almost all values of $x$, up to the finitely many
zeros of the polynomial $q_{L+1}$. ∎

##  5\. Stationary point of determinants for complete graphs

We can now prove our first main result.

######  Theorem 5.1.

Let $M=(V,E,T)$ be a triangulation such that

  1. (1)

the graph is complete, i.e. all vertices are connected by an edge.

  2. (2)

$M$ has no boundary, i.e. each edge is bounded by exactly two triangles.

  3. (3)

the discrete metric $l$ induces angles $\pi/3$ in each triangle.

Consider a smooth family of discrete metrics $l(\varepsilon)$, such that for
$\varepsilon=0$ all angles in the triangles are equal to $\pi/3$. Consider the
corresponding family of cotan-Laplace operators
$\Delta_{\varepsilon}=\Delta_{\text{cot}}(l(\varepsilon))$ and their area
normalized versions $\Delta^{\prime}_{\varepsilon}$ as in Definition 2.3.

  1. (i)

Then $\Delta_{\varepsilon}$ is stationary at $\varepsilon=0$, i.e.

$\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}\log\operatorname{det^{\prime}}\Delta_{\varepsilon}=0.$

  2. (ii)

Assume the total area of triangles for $l(\varepsilon)$ is fixed. Then

$\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}\log\operatorname{det^{\prime}}\Delta^{\prime}_{\varepsilon}=0.$

###### Proof.

The discrete metric $l(\varepsilon)$ defines for each triangle $t\in T$ side
lengths and hence also angles, which we denote by
$(\alpha(\varepsilon),\beta(\varepsilon),\gamma(\varepsilon))$. Since the
angles for $\varepsilon=0$ are all given by $\pi/3$, we can expand

$\displaystyle(\alpha(\varepsilon),\beta(\varepsilon),\gamma(\varepsilon))=(\pi/3,\pi/3,\pi/3)+\varepsilon(t_{\alpha},t_{\beta},t_{\gamma})+O(\varepsilon^{2}),\
\textup{as}\ \varepsilon\to 0,$ (5.1)

where $(t_{\alpha},t_{\beta},t_{\gamma})$ are the infinitesimal angle changes.
Because the sum of angles in each triangle is $\pi$, independently of
$\varepsilon$, we conclude that

$t_{\alpha}+t_{\beta}+t_{\gamma}=\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}\bigl{(}\alpha(\varepsilon)+\beta(\varepsilon)+\gamma(\varepsilon)\bigr{)}=0.$
(5.2)

Let us now consider the family of cotan-Laplace operators
$\Delta_{\varepsilon}$, identified with their corresponding matrices (2.1).
Expanding its weights up to linear order, we get in view of (5.1)

$w_{ij}(\varepsilon)=\cot\alpha_{ij}(\varepsilon)+\cot\beta_{ij}(\varepsilon)=\frac{2}{\sqrt{3}}-\frac{4}{3}(t_{\alpha_{ij}}+t_{\beta_{ij}})\varepsilon+\operatorname{\mathcal{O}}(\varepsilon^{2}).$
(5.3)

Let us abbreviate

$\dot{\Delta}:=\left.\frac{d}{d\varepsilon}\right|_{0}\Delta_{\varepsilon},\quad\dot{w}_{ij}:=\left.\frac{d}{d\varepsilon}\right|_{0}w_{ij}(\varepsilon).$

_Proof of statement $(i)$:_ Observe that by the formula (2.1) and the equation
(5.2)

$\begin{split}\operatorname{Tr}\dot{\Delta}=\sum_{i\in
V}(\dot{\Delta})_{ii}=\sum_{i\in V}\sum\limits_{(i,j)\in
E}\dot{w}_{ij}=2\sum\limits_{(i,j)\in
E}\dot{w}_{ij}=-\frac{8}{3}\sum\limits_{t\in
T}(t_{\alpha}+t_{\beta}+t_{\gamma})=0.\end{split}$ (5.4)

Similarly, we compute for the sum of the off-diagonal terms in the matrix
(2.1)

$\displaystyle\sum_{i\neq j}(\dot{\Delta})_{ij}=-2\sum\limits_{(i,j)\in
E}\dot{w}_{ij}=0.$ (5.5)

One easily sees (here it is essential that the graph is complete, i.e. all
vertices are connected and hence the number of edges emanating from each
vertex is the same) that $\Delta_{0}$ and $(\Delta_{0}+\delta)$ for any
$\delta>0$ are of the same structure as $B$ in Lemma 3.3. Noting that
$(\Delta_{0}+\delta)$ is invertible for $\delta>0$, Lemma 3.3 asserts that for
$(\Delta_{0}+\delta)^{-1}$ all entries on the diagonal are equal, and all
entries away from the diagonal are equal. Hence by Lemma 3.2

$\displaystyle\operatorname{Tr}\dot{\Delta}(\Delta_{0}+\delta)^{-1}\equiv 0.$

Here we should point out, that as long as all weights $w_{ij}>0$ (which is the
case for $\varepsilon$ sufficiently small due to
$w_{ij}(0)=\frac{2}{\sqrt{3}}$) the kernel of $\Delta_{\varepsilon}$ equals
the number of connected components of the graph, i.e. $1$ in this case. Thus,
each $\Delta_{\varepsilon}$ has the same rank and we obtain by Lemma 3.1

$\displaystyle\left.\frac{d}{d\varepsilon}\right|_{0}\log\operatorname{det^{\prime}}\Delta_{\varepsilon}=\lim_{\delta\to
0}\operatorname{Tr}\dot{\Delta}_{0}(\Delta_{0}+\delta)^{-1}=0.$

_Proof of statement $(ii)$:_ We write $A_{i}(\varepsilon)$ for the local area
elements (given by one third of the area of all triangles, cf. Definition 2.3)
at the vertex $i$ for the metric $l(\varepsilon)$. Note that $A_{i}(0)=A$ is
constant, since for $l(\varepsilon=0)$, all triangles have equal sides and by
assumption, the number of edges emanating from each vertex is the same (hence
so is the number of triangles bordering each vertex). Hence we may expand

$\displaystyle A_{i}(\varepsilon)$ $\displaystyle=A+\varepsilon
a_{i}+O(\varepsilon^{2}),$ $\displaystyle\frac{1}{A_{i}(\varepsilon)}$
$\displaystyle=\frac{1}{A}-\frac{\varepsilon
a_{i}}{A^{2}}+O(\varepsilon^{2}),\ \textup{as}\ \varepsilon\to 0.$

Since the total area of the triangles is constant in $\varepsilon$, we find

$\displaystyle\sum_{i\in V}a_{i}=0.$ (5.6)

Let us now consider the family of area normalized cotan-Laplace operators
$\Delta^{\prime}_{\varepsilon}$, identified with their corresponding matrices
(2.1). By the formula (2.1) we obtain (we shall write $n$ for the number of
edges emanating from each vertex and abbreviate by $\dot{\Delta}^{\prime}$ the
derivative of $\Delta^{\prime}_{\varepsilon}$ evaluated at $\varepsilon=0$)

$\begin{split}\operatorname{Tr}\dot{\Delta}^{\prime}=\sum_{i\in
V}(\dot{\Delta}^{\prime})_{ii}&=\frac{1}{A}\sum_{i\in V}\sum\limits_{(i,j)\in
E}\dot{w}_{ij}-\sum_{i\in V}\frac{a_{i}}{A^{2}}\sum\limits_{(i,j)\in
E}w_{ij}(0)\\\ &=\frac{1}{A}\sum_{i\in V}\sum\limits_{(i,j)\in
E}\dot{w}_{ij}-\frac{2n}{A^{2}\sqrt{3}}\sum_{i\in V}a_{i}=0,\end{split}$ (5.7)

where the last equation follows from (5.4) and (5.6). Similarly, we compute
for the sum of the off-diagonal terms in the matrix (2.1)

$\begin{split}\sum_{i\neq
j}(\dot{\Delta}^{\prime})_{ij}&=-2\sum\limits_{(i,j)\in
E}\dot{w}_{ij}+\frac{1}{A^{2}}\sum_{i\in V}a_{i}\sum\limits_{(i,j)\in
E}w_{ij}(0)\\\ &=-2\sum\limits_{(i,j)\in
E}\dot{w}_{ij}+\frac{2n}{A^{2}\sqrt{3}}\sum_{i\in V}a_{i}=0,\end{split}$ (5.8)

where the last equality follows from from (5.5) and (5.6). The rest of the
argument is exactly as in the previous case. ∎

##  6\. Stationary point of determinants for strongly symmetric graphs

We can now generalize Theorem 5.1 to the case of strongly symmetric graphs.

######  Theorem 6.1.

Let $M=(V,E,T)$ be a triangulation such that

  1. (1)

the graph $(V,E)$ is strongly symmetric as in Definition 4.1.

  2. (2)

$M$ has no boundary, i.e. each edge is bounded by exactly two triangles.

  3. (3)

the discrete metric $l$ induces angles $\pi/3$ in each triangle.

Consider a smooth family of discrete metrics $l(\varepsilon)$ such that
$l(0)=l$. Consider the corresponding family of cotan-Laplace operators
$\Delta_{\varepsilon}=\Delta_{\text{cot}}(l(\varepsilon))$ and their area
normalized versions $\Delta^{\prime}_{\varepsilon}$ as in Definition 2.3.

  1. (i)

Then $\Delta_{\varepsilon}$ is stationary at $\varepsilon=0$, i.e.

$\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}\log\operatorname{det^{\prime}}\Delta_{\varepsilon}=0.$

  2. (ii)

Assume the total area of triangles for $l(\varepsilon)$ is fixed. Then

$\left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}\log\operatorname{det^{\prime}}\Delta^{\prime}_{\varepsilon}=0.$

###### Proof.

The argument here differs from the proof in Theorem 5.1 only in the structure
of $(\Delta_{0}+\delta)^{-1}$. For the properties of $\dot{\Delta}$ we still
have (5.4) and (5.5); for the properties of $\dot{\Delta}^{\prime}$ we still
have (5.7) and (5.8), i.e.

$\operatorname{Tr}\dot{\Delta}=\operatorname{Tr}\dot{\Delta}^{\prime}=0,\quad\sum_{i\neq
j}(\dot{\Delta})_{ij}=\sum_{i\neq j}(\dot{\Delta}^{\prime})_{ij}=0.$

By Lemma 4.3, for almost all $\delta>0$ up to a discrete set of values, the
inverse $(\Delta_{0}+\delta)^{-1}$ exists with all entries on the diagonal
being equal, and all entries $(\Delta_{0}+\delta)^{-1}_{ij}$ with $(i,j)\in E$
being equal. Same holds for $\Delta^{\prime}_{0}$. Noting that
$(\Delta_{0})_{ij}=(\Delta^{\prime}_{0})_{ij}=0$ for $(i,j)\notin E$, we
conclude by Lemma 3.2

$\displaystyle\operatorname{Tr}\dot{\Delta}(\Delta_{0}+\delta)^{-1}\equiv
0,\quad\operatorname{Tr}\dot{\Delta}^{\prime}(\Delta^{\prime}_{0}+\delta)^{-1}\equiv
0.$

Note as before in Theorem 5.1, that as long as all weights $w_{ij}>0$ (which
is the case for $\varepsilon$ sufficiently small due to
$w_{ij}(0)=\frac{2}{\sqrt{3}}$) the kernel of $\Delta_{\varepsilon}$ and of
$\Delta^{\prime}_{\varepsilon}$ equals the number of connected components of
the graph, i.e. $1$ in this case. Thus, each
$\Delta_{\varepsilon},\Delta^{\prime}_{\varepsilon}$ has the same rank and we
obtain by Lemma 3.1

$\displaystyle\left.\frac{d}{d\varepsilon}\right|_{0}\log\operatorname{det^{\prime}}\Delta_{\varepsilon}=\lim_{\delta\to
0}\operatorname{Tr}\dot{\Delta}_{0}(\Delta_{0}+\delta)^{-1}=0,$
$\displaystyle\left.\frac{d}{d\varepsilon}\right|_{0}\log\operatorname{det^{\prime}}\Delta^{\prime}_{\varepsilon}=\lim_{\delta\to
0}\operatorname{Tr}\dot{\Delta}^{\prime}_{0}(\Delta^{\prime}_{0}+\delta)^{-1}=0.$

∎

######  Remark 6.2.

As is clear from the proof, Theorem 6.1 actually holds for any triangulation
$M=(V,E,T)$ such that, enumerating the vertices $V=\\{1,\dots,n\\}$, the
inverse matrix $(\Delta_{0}+\delta)^{-1}=(c_{ij})$ satisfies

$\displaystyle\forall i\in V:c_{ii}=x_{0}(\delta),\quad\forall(i,j)\in
E:c_{ij}=x_{1}(\delta).$ (6.1)

for some functions $x_{0}(\delta)$ and $x_{1}(\delta)$, well-defined for
$\delta>0$ at least up to some discrete number of exceptional values. The
notion of strongly symmetric graphs in Definition 4.1 is just one specific
class of graphs that ensures (6.1), which may as well hold in a more general
setting.

##  7\. Numerical illustrations of the stationary points

In order to understand the type of stationary points in Theorems 5.1 and 6.1,
we perform some numerical calculations. We plot the discrete determinants
under variation of only two edge lengths in the following cases:

  1. (1)

the tetrahedron, $4$-point triangulation of the sphere $\mathbb{S}^{2}$ as in
Figure 1;

  2. (2)

the $9$-point triangulation of the torus
$\mathbb{T}=\mathbb{S}^{2}\times\mathbb{S}^{2}$ as in Figure 4.

The plot is presented in Figure 6 using MATLAB to calculate the eigenvalues of
the Laplacian in both cases numerically. The plots strongly suggest that the
cotan-Laplacian attains a local minimum when all edge lengths are equal.

Figure 6. Determinant of the cotan-Laplacian under variation of two side
lengths. Left $-$ in the tetrahedron $4$-point triangulation of
$\mathbb{S}^{2}$; right $-$ in the $9$-point triangulation of
$\mathbb{T}=\mathbb{S}^{2}\times\mathbb{S}^{2}$.

The variation of edge lengths near length $1$ preserves the triangle
inequalities and thus defines a family of triangulations with discrete
metrics. We want to point out that e.g. in the case of the tetrahedron, the
variations can still be realized as convex polyhedrons. Indeed, it is known
that for a triangulation endowed with a discrete metric $l$, the pair of
conditions

  1. (1)

The triangle inequalities hold,

  2. (2)

The Cayley-Menger determinant is positive,

are both necessary and sufficient for the existence of a convex polyhedron
bearing edge lengths induced by $l$, see e.g. [WD09]. The Cayley-Menger
determinant for the tetrahedron is given by the determinant of the matrix

$\begin{pmatrix}0&1&1&1&1\\\ 1&0&l_{12}&l_{13}&l_{14}\\\
1&l_{21}&0&l_{23}&l_{24}\\\ 1&l_{31}&l_{32}&0&l_{34}\\\
1&l_{41}&l_{42}&l_{43}&0\end{pmatrix}.$

For $l_{ij}\equiv 1$ this determinant must be positive since we know that the
uniform tetrahedron exists (in fact it equals $4$). Using the determinant’s
continuity in the matrix components, it immediately follows that the
determinant is positive also in a neighbourhood of uniform edge lengths. Thus,
arbitrary small variations of the uniform metric yield a realizable
tetrahedron.

##  8\. Discussion and outlook

We close the discussion with a list of possible directions worth exploring.

### 1) Different discrete determinants

Our discussion centers around the cotan Laplacian $\Delta_{\text{cot}}$,
motivated by the fact that it appears in the evolution equation for the
curvature $\\{K_{i}\\}$ along discrete Ricci flow, as noted in [SSP08]. There
is a zoo of other meaningful discrete Laplacians to choose from, that may
admit properties better aligned with the Osgood-Philips-Sarnak Theorem, cf.
§1.3.

### 2) Graphs that are not strongly symmetric

An obvious question is whether we can prove a discrete version of the Osgood-
Philips-Sarnak Theorem in a larger class of triangulations, in particular for
those that are not necessarily complete or strongly symmetric.

### 3) Discrete Ricci flow

Ricci flow has been an essential ingredient in the proof of the Osgood-
Philips-Sarnak Theorem. Can we employ the discrete Ricci flow to study
discrete determinants as well? Is the discrete determinant monotone along
discrete Ricci flow? Perhaps it would be necessary to shift the discussion to
the use of circle packing metrics, for which some rigorous theory exists
[CL02].

## References

  * [CL02] B. Chow and F. Luo, _Combinatorial ricci flows on surfaces_ , J. Differential Geom. 63 (2003), no. 1, 97–129.
  * [IzKh20] K. Izyurov and M. Khristoforov, _Asymptotics of the determinant of discrete Laplacians on triangulated and quadrangulated surfaces_ , Comm. Math. Phys. 394 (2022), no. 2, 531–572.
  * [Lee11] J. Lee, _Introduction to topological manifolds_. 2. edition. Grad. Texts in Math., 202. Springer, New York, 2011.
  * [OPS88] B. Osgood, R. Phillips, and P. Sarnak, _Extremals of determinants of laplacians_ , Journal of Functional Analysis 80 (1988), no. 1, 148–211.
  * [SSP08] B. Springborn, P. Schröder, and U. Pinkall, _Conformal equivalence of triangle meshes_ , ACM SIGGRAPH 2008 papers, 2008, pp. 1–11.
  * [WD09] K. Wirth and A. S. Dreiding, _Edge lengths determining tetrahedrons_ , Elem. Math. 64 (2009), no. 4, 160–170.