# Counting Phylogenetic Networks with Few Reticulation Vertices: Galled and
Reticulation-Visible Networks

Yu-Sheng Chang and Michael Fuchs
Department of Mathematical Sciences
National Chengchi University
Taipei 116
Taiwan

###### Abstract

We give exact and asymptotic counting results for the number of galled
networks and reticulation-visible networks with few reticulation vertices. Our
results are obtained with the component graph method, which was introduced by
L. Zhang and his coauthors, and generating function techniques. For galled
networks, we in addition use analytic combinatorics. Moreover, in an appendix,
we consider maximally reticulated reticulation-visible networks and derive
their number, too.

##  1 Introduction and Results

This is the fourth of a series of papers which is concerned with the
enumeration of phylogenetic networks with few reticulation vertices from a
fixed class of phylogenetic networks. In the first three papers, we considered
normal and tree-child networks; see [7, 8, 9]. More precisely, in [7, 9], we
proposed two methods for deriving asymptotic counting results when the number
of reticulation vertices is fixed and the number of leaves tends to infinity.
In [8], we corrected mistakes from the approach from [7] and derived exact
counting formulas. For instance, one of the results of [8] reads as follows.

######  Theorem ([8]).

For the number $N_{\ell,2}$ of normal networks with $\ell$ leaves and two
reticulation vertices,

$N_{\ell,2}=\frac{(3\ell-4)(\ell^{2}+11\ell+6)}{3}(2\ell-1)!!-2^{\ell}(\ell+2)(3\ell-4)\ell!.$

This result solved an open problem from [2] where exact counting results for
tree-child networks were obtained. However, the above result was derived via
generating function techniques, whereas the results from [2] where obtained
with the component graph method. This method as well as generating function
techniques will also play an important role in the current paper.

The main purpose of this paper is to prove exact and asymptotic counting
results for the classes of galled networks and reticulation-visible networks.
We will start with definitions.

######  Definition 1 (Phylogenetic Networks).

A (rooted) phylogenetic network of size $\ell$ is a rooted simple directed
acyclic graph (rooted DAG) whose vertices belong to the following three
categories:

  * (i)

A (unique) root which has indegree $0$ and outdegree $1$;

  * (ii)

Leaves which have indegree $1$ and outdegree $0$ and which are bijectively
labeled with elements from the set $\\{1,\ldots,\ell\\}$;

  * (iii)

Internal vertices which have indegree and outdegree at least $1$ but not both
equal to $1$.

Internal vertices with indegree at least $2$ are called reticulation vertices;
all other internal vertices are called tree vertices. Moreover, a phylogenetic
network is called binary if all internal vertices have total degree equal to
$3$. If not stated otherwise, networks will subsequently always be binary.

Many subclasses of the class of phylogenetic networks have been proposed; see
the recent survey [17] for most of them. The ones of relevance for this paper
are defined next.

To be able to state the definitions, we need two notions: first, a tree cycle
is a pair of two edge disjoint paths from a common tree vertex to a common
reticulation vertex with all remaining vertices being tree vertices; second, a
vertex in a phylogenetic network is called visible, if there exists a leaf
such that any path from the root to the leaf must contain the vertex.

######  Definition 2.

A phylogenetic network is called:

  * (i)

Tree-child network if every non-leaf vertex has at least one child which is
not a reticulation vertex;

  * (ii)

Normal if it is tree-child and has no shortcuts, i.e., the two parents of a
reticulation vertex are not in an ancestor-descendant relationship.

  * (iii)

Galled if each reticulation vertex is in a tree cycle;

  * (iv)

Reticulation-visible if each reticulation vertex is visible.

The set-inclusion relationship of these classes is depicted in Figure 1, where
the class at the bottom is the class of phylogenetic trees which are networks
without reticulation vertices.

Phylogenetic networks Reticulation-visible networks Galled networks Tree-child
networks Normal networks Phylogenetic trees Figure 1: Hasse diagram of the
network classes considered in this paper.

We are interested in enumeration results for these classes. Throughout the
paper, we will use the following notation for the number of networks from a
class with $\ell$ leaves and $k$ reticulation vertices. (The notation for the
number of normal networks already appeared in the above theorem.)

network class | number of networks
---|---
normal networks | $N_{\ell,k}$
tree-child networks | ${\rm TC}_{\ell,k}$
galled networks | ${\rm GN}_{\ell,k}$
reticulation-visible networks | ${\rm RV}_{\ell,k}$
phylogenetic networks | ${\rm PN}_{\ell,k}$

The following asymptotic result for these numbers is known. For fixed $k$,

${\rm PN}_{\ell,k}\sim{\rm TC}_{\ell,k}\sim{\rm
N}_{\ell,k}\sim\frac{2^{k-1}\sqrt{2}}{k!}\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+2k-1},\qquad(\ell\rightarrow\infty).$

The first asymptotic equivalence was proved in [18] and the second and third
in [7, 8] (without the closed-form expression for the leading constant) and in
[9] (with the closed-form expression for the leading constant). An easy
consequence of these results is the following.

######  Corollary 1.

For fixed $k$,

${\rm
RV}_{\ell,k}\sim\frac{2^{k-1}\sqrt{2}}{k!}\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+2k-1},\qquad(\ell\rightarrow\infty).$

Thus, for fixed $k$, the asymptotics of the number of networks for all classes
from Definition 2 is known except for the class of galled networks. Here,
somehow surprisingly, again the same result holds.

######  Theorem 2.

For fixed $k$,

${\rm
GN}_{\ell,k}\sim\frac{2^{k-1}\sqrt{2}}{k!}\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+2k-1},\qquad(\ell\rightarrow\infty).$

We will prove this result below. Our method will also allow us to obtain
closed-form expressions for ${\rm GN}_{\ell,k}$ for small $k$. We state the
results for $k=2$ and $k=3$.

######  Theorem 3.

We have,

${\rm
GN}_{\ell,2}=\frac{6\ell^{4}+31\ell^{3}+30\ell^{2}-7\ell-9}{3}(2\ell-3)!!-2^{\ell-2}(7\ell+10)(\ell+1)!$

and

$\displaystyle{\rm GN}_{\ell,3}=$
$\displaystyle\frac{140\ell^{6}+3184\ell^{5}+17195\ell^{4}+34125\ell^{3}+19475\ell^{2}-8599\ell-6090}{105}(2\ell-3)!!$
$\displaystyle\qquad\qquad-2^{\ell-5}\frac{225\ell^{3}+2045\ell^{2}+5878\ell+5448}{3}(\ell+1)!.$

######  Remark 1.

The formula for $k=2$ already appeared in [2] (with typos which we have
corrected here).

Moreover, we will also give similar formulas for ${\rm RV}_{\ell,k}$ for small
$k$.

######  Theorem 4.

We have,

${\rm
RV}_{\ell,2}=\frac{6\ell^{4}+7\ell^{3}+6\ell^{2}-\ell-3}{3}(2\ell-3)!!-2^{\ell-1}(2\ell^{2}+2\ell+1)\ell!$

and

$\displaystyle{\rm RV}_{\ell,3}=$
$\displaystyle\frac{4\ell^{6}+20\ell^{5}+33\ell^{4}-32\ell^{3}-76\ell^{2}+12\ell+12}{3}(2\ell-3)!!$
$\displaystyle\qquad-2^{\ell-4}\frac{48\ell^{4}+175\ell^{3}+99\ell^{2}-262\ell-264}{3}\ell!.$

######  Remark 2.

Note that for $k=0$, all the numbers coincide, i.e.,

${\rm PN}_{\ell,0}={\rm RV}_{\ell,0}={\rm GN}_{\ell,0}={\rm TC}_{\ell,0}={\rm
N}_{\ell,0}=(2\ell-3)!!,$

where the last number is the (well-known) number of phylogenetic trees with
$\ell$ leaves. Also, for $k=1$, all the numbers, except the number of normal
network, coincide:

${\rm PN}_{\ell,1}={\rm RV}_{\ell,1}={\rm GN}_{\ell,1}={\rm
TC}_{\ell,1}=\ell(2\ell-1)!!-2^{\ell-1}\ell!,$

where the last result was derived, e.g., in [2]. ($N_{\ell,1}$ is known as
well; see, e.g., [20].)

Finally, we will consider maximal reticulated networks. Here, the following
sharp upper bounds for the number of reticulation vertices $k$ have been
proved.

network class | maximal number of reticulation vertices
---|---
normal networks | $\ell-2$
tree-child networks | $\ell-1$
galled networks | $2\ell-2$
reticulation-visible networks | $3\ell-3$

The result for normal networks first appeared in [19]; the result for tree-
child networks was discovered by many authors (and is, in fact, easy to
prove). For galled networks, the optimal upper bound was proved, e.g., in [13,
16]; see also [11]. Finally, for reticulation-visible networks, in [12], the
upper bound $4\ell-4$ was established and then this upper bound was reduced to
the optimal one independently in [1] and [16]. Here, we will give a simplified
proof which in addition also gives the number of maximal reticulated
reticulation-visible networks.

######  Theorem 5.

The maximum number of reticulation vertices in a reticulation-visible networks
with $\ell$ leaves equals $3\ell-3$. Moreover,

${\rm RV}_{\ell,3\ell-3}={\rm
TC}_{\ell,\ell-1}=\Theta\left(\ell^{-2/3}e^{a_{1}(3\ell)^{1/3}}\left(\frac{12}{e^{2}}\right)^{\ell}\ell^{2\ell}\right),$

where $a_{1}$ is the largest root of the Airy function of the first kind.

######  Remark 3.

The asymptotic result follows from the main result in [10].

We will conclude the introduction by an outline of the paper. The main method
of proof will be the component graph method which was introduced by Zhang and
his coauthors. We will recall this method and some related results in the next
section. In Section 3, we will then combine it with generating function
techniques and tools from analytic combinatorics to prove Theorem 2 and
Theorem 3 for galled networks. Section 4 will contain the proof of Theorem 4
for reticulation-visible networks. Theorem 5 is not really concerned with
phylogenetic networks with few reticulation vertices and therefore does not
really fall into the scope of this paper. Therefore, we will prove it in the
appendix. The paper will be concluded with some final remarks in Section 5.

##  2 The Component Graph Method

The component graph method was introduced by Zhang and used by him and his
coauthors to solve algorithmic problems for phylogenetic networks; see [2, 13,
14, 15]. For instance, it was used to algorithmically solve the counting
problem for tree-child networks in [2] and galled networks in [14].

The central object of the method are component graphs. First, for a given
phylogenetic network, its tree components are obtained by removing the two
incoming edges of every reticulation vertex. (These edges are called
reticulation edges.) Using these objects, component graphs are defined as
follows.

######  Definition 3 (Component Graph).

The component graph of a phylogenetic network $N$ has a vertex for each tree-
component. These vertices are then connected via edges according to how the
tree-components are connected in $N$ via the reticulation edges. Finally, we
attach the leaves (with their labels) in the tree-components to their vertices
in the component graph except when the tree-component has only one leaf and
corresponds to a terminal vertex in the component graph with incoming edges a
double edge, in which case we use the label of the leaf to label the terminal
vertex.

See Figures 2 for a phylogenetic network and its component graph. Note that we
have replaced double edges by single edges but have indicated that they are
double edges by placing arrows on them.

12345678 $N$

13456872 $C(N)$

Figure 2: Left: A reticulation-visible network with $8$ leaves and $8$
reticulation vertices (with the tree-components highlighted). Right: The
component graph of the network. Note that it is tree-child network.

123456789101112 $N$

234578911121061 $C(N)$

Figure 3: Left: A galled network with $12$ leaves and $9$ reticulation
vertices (with the tree-components highlighted). Right: The component graph of
the network. Note that it is a phylogenetic tree.

If $N$ is a network, we denote its component graph by $C(N)$. The component
graph can be seen as a compression of the network. We also use the notation
$\tilde{C}(N)$ to denote the component graph with the arrows on single edges
removed.

Note that the phylogenetic network in Figure 2 is a reticulation-visible
network and its component graph is a (non-binary) tree-child network. This, in
fact, is a general phenomenon.

######  Theorem ([15]).

Let $N$ be a phylogenetic network.

  * (i)

$N$ is galled if and only if $\tilde{C}(N)$ is a (not necessarily binary)
phylogenetic tree.

  * (ii

$N$ is reticulation-visible if and only if $\tilde{C}(N)$ is a (not
necessarily binary) tree-child network with all vertices of indegree at most
$2$ and no reticulation vertex just followed by one tree vertex.

See Figure 3 for an illustration of part (i) of the above result.

In order to generate networks, we start from the the component graphs and
decompress them. For this, the notion of one-component phylogenetic networks
is useful.

######  Definition 4.

A phylogenetic network is called a one-component phylogenetic network if every
reticulation vertex is directly followed by a leaf.

In other words, a network is one-component if it has only one non-trivial
tree-component.

One-component galled networks were counted in [14]. Denote by ${\rm
OGN}_{\ell,k}$ the number of one-component galled networks with $\ell$ leaves
and $k$ reticulation vertices and by $M_{\ell,k}$ those one-component galled
networks whose leaves below the reticulation vertices are labeled by
$\\{1,\ldots,k\\}$. Then, the following result was proved in [14]. (Note that
part (i) is trivial.)

######  Proposition ([14]).

  * (i)

We have,

${\rm OGN}_{\ell,k}=\binom{\ell}{k}M_{\ell,k}.$

  * (ii)

For $2\leq k\leq\ell$,

$\displaystyle M_{\ell,k}=(\ell+k-2)$ $\displaystyle
M_{\ell,k-1}+(k-1)M_{\ell,k-2}$ $\displaystyle+\frac{1}{2}\sum_{1\leq d\leq
k-1}\binom{k-1}{d}(2d-1)!!\left(M_{\ell-d,k-1-d}-M_{\ell+1-d,k-1-d}\right),$
(1)

initial values $M_{\ell,0}=(2\ell-3)!!$ and $M_{\ell,1}=(\ell-1)(2\ell-3)!!$.

Now, consider the component graph of a galled network (with at least $2$
vertices). In order to decompress it, first pick a one-component galled
network ${\rm ON}$ for the root $r$ of the component graph, where the number
of leaves, say $\ell_{r}$, of ${\rm ON}$ is the outdegree of $r$ and the
number of reticulation vertices, say $k_{r}$, is the number of arrows on the
outgoing edges of $r$. Moreover, let the leaves below the reticulation
vertices in ${\rm ON}$ have labels $\\{1,\ldots,k_{r}\\}$. Next, remove $r$
from the component graph which gives a forest of $\ell_{r}$ trees. The trees
which have been attached by edges with arrows to $r$ go to the leaves below
reticulation vertices in ${\rm ON}$ and the remaining trees (which consist of
just one labeled vertex) are used the relabel the remaining leaves of ${\rm
ON}$ in an order consistent way (i.e., the smallest goes to $k_{r}+1$, the
second smallest to $k_{r}+2$, etc.). Moreover, for the trees which go to the
leaves below reticulation vertices, the order in which they are attached is
the increasing order of their smallest leaf label (i.e., the one with the
smallest leaf label goes to $1$, the one with the second smallest leaf label
goes to $2$, etc.). Then, continue with the non-leave vertices of the trees in
a recursive way. This gives, by picking all possible one-component networks in
each step, all possible galled networks whose component graph is the given
component graph. Moreover, if the given component graph has no arrows (which
for galled networks means it is just a phylogenetic tree), then arrows have to
be placed on edges leading to non-leaf vertices, whereas on the pendant edges,
we can freely decide whether we want to place an arrow or not.

Overall, this procedure gives the following result from [14] for the number of
galled networks with $\ell$ leaves, denoted by ${\rm GN}_{\ell}$.

######  Theorem ([14]).

For the number ${\rm GN}_{\ell}$ of galled networks with $\ell$ leaves,

${\rm GN}_{\ell}=\sum_{{\mathcal{T}}}\prod_{v}\sum_{j=0}^{c_{\rm
lf}(v)}\binom{c_{\rm lf}(v)}{j}M_{c(v),c_{\rm nlf}(v)+j},$ (2)

where the first sum runs over all (not necessarily binary) phylogenetic trees
${\mathcal{T}}$, the product runs over all internal vertices $v$ of
${\mathcal{T}}$, $c(v)$ is the outdegree of $v$ and $c_{{\rm lf}}(v)$ resp.
$c_{{\rm nlf}}(v)$ are the number of leaves resp. non-leaves amongst the
children of $v$.

By keeping track of the number of arrows, ${\rm GN}_{\ell,k}$ can be computed
as well.

Next, a similar procedure can be used for reticulation-visible networks, too.
First, observe that the classes of one-component reticulation-visible networks
and one-component galled networks coincide. Thus, we can again work with
$M_{\ell,k}$ which satisfies (1).

Now, consider the component graph of a reticulation-visible network (which by
the above result is a particular tree-child network; also, assume that there
are arrows on the edges). The decompressing works as for a galled networks,
with the only difference of how the root $r$ is replaced by a one-component
network. More precisely, in which order do we attach the children of $r$ with
and without arrows to the leaves of the one-component network which replaces
$r$? (The children attached with edges having arrows to $r$ go to the leaves
below reticulation vertices, the others not.) Here, the tree-child property
comes into play. It implies that that for every vertex, there exists a set of
leaves which can be only reached from this vertex. All children of $r$ have
such a set and we can order these sets (and consequently the children of $r$)
according to their smallest elements. This is then used to attach the children
of $r$ to the one-component network which replaces $r$.

Overall, this gives the following result for the number of reticulation-
visible networks with $\ell$ leaves, denoted by ${\rm RV}_{\ell}$, where we
again start from component graphs without arrows.

######  Theorem 6.

For the number ${\rm RV}_{\ell}$ of reticulation-visible networks with $\ell$
leaves,

${\rm RV}_{\ell}=\sum_{{\mathcal{TC}}}\prod_{v}\sum_{j=0}^{c_{\rm
lf}(v)}\binom{c_{\rm lf}(v)}{j}M_{c(v),c_{1}(v)+j},$ (3)

where the first sum runs over all (not necessarily binary) tree-child networks
with all vertices of indegree at most $2$ and no reticulation vertex just
followed by one tree vertex, the product runs over all internal vertices $v$
of the tree-child network, $c(v)$ is the outdegree of $v$, $c_{\rm lf}(v)$ is
the number of leaves below $v$, and $c_{1}(v)$ is the number of children of
$v$ which are not leaves and have indegree $1$.

Again, by keeping track of arrows, we can compute ${\rm RV}_{\ell,k}$. We will
do this in Section 4 to derive formulas for small values of $k$ and any
$\ell$.

##  3 Galled Networks

In this section, we will use the component graph method to count (both exactly
and asymptotically) galled networks with a small number of reticulation
vertices. In addition, we will use generating function techniques and the
method of singularity analysis which is described in Chapter VI of [6]. (We
will use below some of the notation from that chapter.)

Since the component graph method is based on one-component galled networks, we
start by analyzing their number. First, from the initial values and (1),
$M_{\ell,k}$ for small values of $k$ equals

$\displaystyle M_{\ell,0}$ $\displaystyle=(2\ell-3)!!,$ $\displaystyle
M_{\ell,1}$ $\displaystyle=(\ell-1)(2\ell-3)!!,$ $\displaystyle M_{\ell,2}$
$\displaystyle=(2\ell-1)(\ell-1)^{2}(2\ell-5)!!.$

From this, we observe the following pattern.

######  Lemma 7.

For fixed $k\geq 1$,

$M_{\ell,k}=p_{k}(\ell)(2(\ell-k)-3)!!,\qquad(\ell\geq k),$

where $p_{k}(\ell)$ is a polynomial of degree $2k$ with leading coefficient
$2^{k}$.

###### Proof.

This follows by induction on $k$. First, the claim holds for $k=1,2,3$ and
also for $k=0$ when $\ell\geq 1$. (This is why we used $(2(\ell-k)-3)!!$
instead of $(2(\ell-k+1)-3)!!$.)

Assume now that it holds for $2\leq k^{\prime}<k$. Then, by (1), we see that
it also holds for $k$ where the highest order leading term comes from the
first term on the right-hand side of (1), namely, it is the highest order
leading term of $p_{k-1}(\ell)$ multiplied by $2\ell^{2}$. This proves the
claim.

Next, we consider the exponential generating function of $M_{\ell,k}$:

$M_{k}(z)=\sum_{\ell\geq k}M_{\ell,k}\frac{z^{\ell}}{\ell!}.$

From the expressions for $M_{\ell,k}$ for $k=0,1,2$, we obtain that

$M_{0}(z)=1-\sqrt{1-2z},\qquad
M_{1}(z)=\frac{(1-\sqrt{1-2z})^{2}}{2\sqrt{1-2z}}$

and

$M_{2}(z)=\frac{(4z^{2}-7z+4+\sqrt{1-2z})(1-\sqrt{1-2z})^{2}}{6(1-2z)^{3/2}}.$

Also, we have the following asymptotic result for $M_{k}(z)$ for all $k\geq
1$, where we use the notion of $\Delta$-analyticity of a function $f(z)$ at
$z_{0}$, i.e., $f(z)$ is an analytic function in a domain of the form

$\Delta:=\\{z\ :\ |z|<r,\ |\arg(z-z_{0})|>\phi\\},$

for some $r>|z_{0}|$ and $0<\phi<\pi/2$.

######  Lemma 8.

For $k\geq 1$, $M_{k}(z)$ is $\Delta$-analytic for some $\Delta$-domain at
$1/2$ and satisfies, as $z\rightarrow 1/2$, in the $\Delta$-domain:

$M_{k}(z)\sim\frac{(2k-3)!!}{2^{k}(1-2z)^{k-1/2}}.$

###### Proof.

Note that

$P_{k}(z):=\sum_{\ell\geq
k}(2(\ell-k)-3)!!\frac{z^{\ell}}{\ell!}=\sum_{\ell\geq
0}(2\ell-3)!!\frac{z^{\ell}}{\ell!}=2-\sqrt{1-2z}.$

From Lemma 7, we see that the function $M_{k}(z)$ is built from $P_{k}(z)$ as
a linear combination of

$D^{j}P_{k}(z),$

where $0\leq j\leq 2k$ and $D^{j}$ is the $j$-th iteration of
$D:=z\frac{d}{dz}$. This shows that $M_{k}(z)$ is $\Delta$-analytic with
singularity expansion, as $z\rightarrow 1/2$,

$M_{k}(z)\sim 2^{k}D^{2k}P_{k}(z)=\frac{(2k-3)!!}{2^{k}(1-2z)^{k-1/2}}$

which is the claim.

We also need the following closely related generating function

$F_{k}(z)=\sum_{\ell\geq 0}M_{\ell+k,k}\frac{z^{\ell}}{\ell!}$

which satisfies the following result.

######  Corollary 9.

For $k\geq 1$, $F_{k}(z)$ is $\Delta$-analytic for some $\Delta$-domain at
$1/2$ and satisfies, as $z\rightarrow 1/2$ in the $\Delta$-domain:

$F_{k}(z)\sim\frac{(4k-3)!!}{2^{k}(1-2z)^{2k-1/2}}.$ (4)

###### Proof.

Note that

$F_{k}(z)=\frac{d^{k}}{dz^{k}}M_{k}(z).$ (5)

Thus, the result follows from the closure properties of singularity analysis;
see Section VI.6 in [6]

######  Remark 4.

Note that $k=0$ has a different singularity expansion since
$F_{0}(z)=1-\sqrt{1-2z}$. Also, from the above results for $M_{k}(z)$ and (5):

$F_{1}(z)=\frac{z}{(1-2z)^{3/2}},\qquad
F_{2}(z)=\frac{3-z+7z^{2}-4z^{3}}{(1-2z)^{7/2}}.$

We now consider general galled networks which are built from one-component
galled networks and component graphs. This was described in detail in Section
2 and the recursive method there can be translated into generating functions
(because the component graphs are trees). Since we are interested in
reticulation vertices, we need to keep track of them. We therefore consider
the generating function

$G(z,v):=\sum_{\ell\geq 0}\sum_{k\geq 0}{\rm
GN}_{\ell,k}\frac{z^{\ell}}{\ell!}v^{k}.$

Then, we have the following result.

######  Proposition 10.

We have,

$G(z,v)=\sum_{j\geq 0}F_{j}(z)\frac{(vG(z,v))^{j}}{j!}.$ (6)

###### Proof.

According to the decompressing procedure of component graphs of galled
networks described in the last section, every galled network is obtained from
a one-component galled network (this is the term $F_{j}(z)$) with the leaves
below the reticulation vertices replaced by an (unordered) sequence of galled
networks (this is the term $G(z,v)^{j}/j!$; $v^{j}$ keeps track of the
reticulation vertices). (Note that the relabeling of the leaves of the one-
component galled network which replaces the root of the component graph
described in the previous section is “automatically” obtained from the product
of the exponential generating functions which corresponds in combinatorics to
the product of labeled combinatorial classes; see, e.g., Chapter II in [6]
were this is explained in detail.)

The exponential generating function for the number of galled networks with $k$
reticulation vertices, i.e.,

$E_{k}(z):=\sum_{\ell\geq k}{\rm GN}_{\ell,k}\frac{z^{\ell}}{\ell!}$

is obtained from $G(z,v)$ by partial differentiation

$E_{k}(z)=\frac{1}{k!}\frac{\partial^{k}}{\partial v^{k}}G(z,v)\Big{|}_{v=0}.$

From Proposition 10, we obtain a recurrence.

######  Lemma 11.

For $k\geq 1$,

$E_{k}(z)=\sum_{j=1}^{k}\frac{F_{j}(z)}{j!}\sum_{\ell_{1}+\cdots+\ell_{j}=k-j}E_{\ell_{1}}(z)\cdots
E_{\ell_{j}}(z).$ (7)

###### Proof.

Differentiating (6) $k$-times gives

$\displaystyle E_{k}(z)$
$\displaystyle=\frac{1}{k!}\sum_{j=1}^{k}\frac{F_{j}(z)}{j!}\frac{d^{k}}{dv^{k}}\left(vG(z,v)\right)^{j}\Big{|}_{v=0}$
$\displaystyle=\frac{1}{k!}\sum_{j=1}^{k}\frac{F_{j}(z)}{j!}\binom{k}{j}j!\frac{d^{k-j}}{dz^{k-j}}G(z,v)^{j}$
$\displaystyle=\sum_{j=1}^{k}\frac{F_{j}(z)}{j!(k-j)!}\sum_{\ell_{1}+\cdots+\ell_{j}=k-j}\binom{k-j}{\ell_{1}!\cdots\ell_{j}!}\ell_{1}!E_{\ell_{1}}(z)\cdots\ell_{j}!E_{\ell_{j}}(z).$

The claim is obtained by simplifying this.

We have now everything ready to prove Theorem 3.

###### Proof of Theorem 3..

Note that $E_{0}(z)=1-\sqrt{1-2z}$. Then, from (7) and Remark 4

$E_{1}(z)=F_{1}(z)E_{0}(z)=\frac{z(1-\sqrt{1-2z})}{(1-2z)^{3/2}}.$

Moreover, by using (7) and Remark 4 once again,

$\displaystyle E_{2}(z)$
$\displaystyle=F_{1}(z)E_{1}(z)+\frac{F_{2}(z)E_{0}(z)^{2}}{2}$
$\displaystyle=\frac{12z^{4}-18z^{3}+17z^{2}-36z+21+(12z^{3}-10z^{2}+15z-21)\sqrt{1-2z}}{3(1-2z)^{7/2}}.$

Extracting coefficients gives the claimed result for ${\rm GN}_{\ell,2}$.

As for ${\rm GN}_{\ell,3}$, the same method can be used, only the resulting
computation is more tedious (and therefore best done with mathematical
software, e.g., Maple).

What is left is to prove the asymptotic counting result for ${\rm
GN}_{\ell,k}$ from Theorem 2. This will be follow from the following
asymptotic result for $E_{k}(z)$ which is obtained from (7) and induction.

######  Proposition 12.

For $k\geq 1$, $E_{k}(z)$ is $\Delta$-analytic for some $\Delta$-domain at
$1/2$ and satisfies, as $z\rightarrow 1/2$ in the $\Delta$-domain:

$E_{k}(z)\sim\frac{(4k-3)!!}{k!2^{k}(1-2z)^{2k-1/2}}.$

###### Proof.

Note that

$E_{0}(z)=F_{0}(z)=1-\sqrt{1-2z}\sim 1.$

Thus, if $k=0$ is included in the claim, then the power of $1-2z$ in the
denominator is $\max\\{2k-1/2,0\\}$.

We use induction on $k$. For $k=1$, the claim holds since $E_{1}(z)\sim
F_{1}(z)$ which has the desired form by (4). Thus, we can assume that the
claim holds for $k^{\prime}<k$. We need to prove it for $k$. Plugging the
induction hypothesis into (7) and using (4), we obtain that for the terms
inside the double sum of (7):

$F_{j}(z)E_{\ell_{1}}(z)\cdots E_{\ell_{j}}(z)\sim
c(1-2z)^{-(2j-1/2+\max\\{2\ell_{1}-1/2,0\\}+\cdots+\max\\{2\ell_{j}-1/2,0\\})},$

where $c$ is a suitable constant and $\ell_{1}+\cdots+\ell_{j}=k-j$. The term
inside the bracket is maximized if and only if $j=k$ and thus
$\ell_{1}=\cdots=\ell_{k}=0$. This shows that

$E_{k}(z)\sim\frac{F_{k}(z)}{k!}$

which by (4) gives the claim.

Now, we can prove Theorem 2 for $k\geq 1$.

###### Proof of Theorem 2..

From Proposition 7 and the transfer theorems of singularity analysis (see
Chapter VI in [6]):

${\rm
GN}_{\ell,k}=\ell![z^{\ell}]E_{k}(z)\sim\ell!\frac{(4k-3)!!}{k!2^{k}\Gamma(2k-1/2)}2^{\ell}\ell^{2k-3/2}.$

Note that

$\Gamma(2k-1/2)=2^{-2k+1}(4k-3)!!\sqrt{\pi}.$

Plugging this into the above expression and using Stirling’s formula gives the
claimed result.

######  Remark 5.

It is easily verified that Theorem 2 holds for $k=0$, too.

##  4 Reticulation-Visible Networks

In this section, we will prove the formulas for the number of reticulation-
visible networks with $\ell$ leaves and $k$ reticulation vertices with $k=2$
and $k=3$; see Theorem 4.

Let $N$ be a reticulation-visible network. We remove the leaves and their
pendant edges from $C(N)$ except those whose pendant edge has an arrow on it.
Then, the resulting component graph is a (unlabeled) rooted simple DAG with
every non-root vertex of indegree $2$. (Note that an edge with an arrow on it
is actually a double edge and thus counts as two edges.) Moreover, this DAG
has exactly $k+1$ vertices. For $k=2$ and $k=3$, all these DAGs are listed in
Figure 8 in [2]; see also Figure 4 and Figure 5 below.

Using these DAGs and the decompression procedure explained in Section 2, we
can derive the exponential generating function of ${\rm RV}_{\ell,k}$.

######  Proposition 13.

Let $\mathcal{D}_{m}$ be the set of (unlabelled) rooted DAGs with $m$ vertices
in which non-root vertices have indegree $2$ and double edges between two
vertices are allowed. Then, for given $k$,

$\sum_{\ell\geq 1}\,{\rm
RV}_{\ell,k}\,\frac{z^{\ell}}{\ell!}=\sum_{G\in\mathcal{D}_{k+1}}\,\frac{1}{{\rm
m}(G)}\prod_{v\in
G}\,\sum_{\ell\geq\ell_{0}}\,M_{\ell+c(v),c_{1}(v)}\,\frac{z^{\ell}}{\ell!},$
(8)

where ${\rm m}(G)$ counts symmetries in $G$, the product runs over all
vertices in $G$, and the final sum on the right-hand side is the generating
function with respect to the number of labeled leaves which are attached to
$v$, where $c(v)$ is the outdegree of $v$ in $G$, $c_{1}(v)$ is the number of
children of $v$ with arrow on their edges, and $\ell_{0}$ is $0$ or $1$
according to whether $c_{1}(v)>0$ or $c_{1}(v)=0$, respectively.

$a_{1}$$a_{2}$$a_{3}$ $A$

$b_{1}$$b_{2}$$b_{3}$ $B$

$c_{1}$$c_{2}$$c_{3}$ C

Figure 4: The $3$ DAGs from the set $\mathcal{D}_{3}$ where for convenience,
we have labeled the vertices. Call the reticulation vertices of the
decompressed reticulation-visible networks $r_{1}$ and $r_{2}$. Then, $A$
gives all the networks where $r_{1}$ and $r_{2}$ are not in an ancestor-
descendant relationship; $B$ gives all networks where $r_{1}$ is above $r_{2}$
but they are not in a tree-cycle; and $C$ gives all networks where $r_{1}$ is
in the tree cycle of $r_{2}$.

Now, to find ${\rm RV}_{\ell,2}$, we start from the DAGs in the set
$\mathcal{D}_{3}$ which are listed in Figure 4. Note that $m(A)=2$ (due to the
symmetry about the root)$,m(B)=m(C)=1$, and for each vertex $v$, the
exponential generating function $f_{v}(z)$ from Proposition 13 equals (here,
$X=\sqrt{1-2z}$):

$\displaystyle f_{a_{1}}(z)$ $\displaystyle=F_{2}(z)=\sum_{\ell\geq
0}M_{\ell+2,2}\,\frac{z^{n}}{n!}=\frac{15}{4}\,X^{-7}-\frac{3}{2}\,X^{-5}+\frac{1}{4}\,X^{-3}+\frac{1}{2}\,X^{-1};$
$\displaystyle f_{a_{2}}(z)$
$\displaystyle=f_{a_{3}}(z)=f_{b_{3}}(z)=f_{c_{3}}(z)=F_{0}(z)=\sum_{\ell\geq
1}M_{\ell,0}\,\frac{z^{\ell}}{\ell!}=1-X;$ $\displaystyle f_{b_{1}}(z)$
$\displaystyle=f_{b_{2}}(z)=F_{1}(z)=\sum_{\ell\geq
0}M_{\ell+1,1}\,\frac{z^{\ell}}{\ell!}=\frac{1}{2}\,X^{-3}-\frac{1}{2}\,X^{-1};$
$\displaystyle f_{c_{1}}(z)$ $\displaystyle=\sum_{\ell\geq
0}M_{\ell+2,1}\,\frac{z^{\ell}}{\ell!}=\frac{3}{2}\,X^{-5}-\frac{1}{2}\,X^{-3};$
$\displaystyle f_{c_{2}}(z)$ $\displaystyle=\sum_{\ell\geq
1}M_{\ell+1,0}\,\frac{z^{\ell}}{\ell!}=X^{-1}-1,$

Thus, from Proposition 13, we have for the exponential generating function of
${\rm RV}_{\ell,2}$,

$\frac{(3-z+7z^{2}-4z^{3})(1-z-\sqrt{1-2z})}{(1-2z)^{7/2}}=\frac{(1-X)^{2}(15-6X^{2}+X^{4}+2X^{6})}{8X^{7}}.$

What is left is to extract coefficients which can be done with the following
lemma.

######  Lemma 14.

We have, for $n$ large enough,

$[z^{n}]X^{d}=\left\\{\begin{array}[]{cl}\displaystyle 0,&\text{if }d\geq
0\text{ and }d\text{ is even, }k:=\frac{d}{2};\\\
\displaystyle(-1)^{k+1}(2k+1)!!\frac{(2n-2k-3)!!}{n!},&\text{if }d\geq 0\text{
and }d\text{ is odd, }k:=\frac{d-1}{2};\\\ \displaystyle
2^{n}\,\binom{n+k-1}{k-1},&\text{if }d<0\text{ and }d\text{ is even,
}k:=-\frac{d}{2};\\\
\displaystyle\frac{1}{(2k-3)!!}\frac{(2n+2k-3)!!}{n!},&\text{if }d<0\text{ and
}d\text{ is odd, }k:=-\frac{d-1}{2}.\\\ \end{array}\right.$

Applying the lemma gives the following result.

${\rm
RV}_{\ell,2}=\frac{6\ell^{4}+7\ell^{3}+6\ell^{2}-\ell-3}{3}(2\ell-3)!!-2^{\ell-1}(2\ell^{2}+2\ell+1)\ell!.$

$A_{1}$

$A_{2}$

$A_{3}$

$A_{4}$

$B_{1}$

$B_{2}$

$B_{3}$

$B_{4}$

$B_{5}$

$B_{6}$

$B_{7}$

$B_{8}$

$B_{9}$

Figure 5: The $13$ DAGs from the set $\mathcal{D}_{4}$.

We next consider $k=3$. Here the set $\mathcal{D}_{4}$ has $13$ elements; see
Figure 5. Note that these DAGs fall into two types: the $A_{j}$’s are tree
structures and the $B_{j}$’s are not. (The latter generate the reticulation-
visible networks which are not galled networks.) Next, we note that
$m(A_{1})=6,m(A_{2})=m(B_{1})=m(B_{4})=2$ and the values are $1$ in all other
cases. Thus, by Proposition 13, we obtain for the exponential generating
function $f_{A}(z)$ arising by the $A_{j}$’s,

$f_{A}(z)=\frac{4z^{3}(29+12z+29z^{2}-37z^{3}+36z^{4}-14z^{5})}{(1-2z)^{11/2}\,(1+\sqrt{1-2z})^{3}}\\\
+\frac{6z^{3}(3-z+7z^{2}-4z^{3})}{(1-2z)^{5}\,(1+\sqrt{1-2z})^{2}}+\frac{2z^{4}}{(1-2z)^{9/2}\,(1+\sqrt{1-2z})},$

and for the exponential generating function $f_{B}(z)$ arising from the
$B_{j}$’s,

$f_{B}(z)=\frac{(1-X)^{2}(258-105X-153X^{2}-16X^{3}+26X^{4}+7X^{5}+3X^{6}-2X^{7}-2X^{8})}{8\,X^{10}}.$

Then, by extracting coefficients of $f_{A}(z)+f_{B}(z)$ with Lemma 14,

$\displaystyle{\rm RV}_{\ell,3}=$
$\displaystyle\frac{4\ell^{6}+20\ell^{5}+33\ell^{4}-32\ell^{3}-76\ell^{2}+12\ell+12}{3}(2\ell-3)!!$
$\displaystyle\qquad-2^{\ell-4}\frac{48\ell^{4}+175\ell^{3}+99\ell^{2}-262\ell-264}{3}\ell!.$

##  5 Conclusion

In this paper, we derived exact and asymptotic counting results for the number
of galled networks and reticulation-visible networks with a fixed number of
reticulation vertices. For galled networks, we proposed a generating function
approach which is based on the component graph method by Zhang and his
coauthors. The exact results followed from it by coefficient extraction.
Moreover, the asymptotic result was derived from it by methods from analytic
combinatorics. For reticulation-visible networks, we also used the component
graph method and generating functions to derive their exact counts. Moreover,
the asymptotic result followed from previous work.

Thus, an asymptotic counting result for fixed $k$ is now known for all network
classes from Figure 1; in fact, the asymptotic main term is the same for all
these classes. This is slightly surprising for galled networks, since they are
on a different “branch” in the diagram in Figure 1. This suggests to consider
phylogenetic networks which are both galled and satisfy the tree-child
condition. This class of galled tree-child networks is also interesting from a
theoretical point of view since very different methods have been used to
determine the (asymptotic) numbers of tree-child and galled networks; see [10,
11]. We will do this in the forthcoming work [5].

All the results in this paper were proved for fixed $k$. How about the number
of networks if one sums over all possible values of $k$? The first order
asymptotics of this number for galled networks is known and was obtained in
[11]. In fact, the latter paper also uses the component graph method. More
precisely, the asymptotics was deduced from (2). The analysis proceeded in two
steps: first, the number of one-component galled networks was asymptotically
studied by using the recurrence (1). It turned out that $M_{\ell,k}$ has its
maximum at $k=\ell$ (which is also the largest possible value of $k$) and
satisfies a Poisson law. Then, in the second step, (2) was used to find the
asymptotics of ${\rm GN}_{\ell}$. This was possible because the first sum is
over the set of (not necessarily binary) phylogenetic trees and the authors in
[11] found the trees whose contributions dominate the asymptotics.

In view of (3) does this approach also work for reticulation-visible networks?
First, we can skip the first step because, as mentioned in Section 2, the
class of one-component galled networks and one-component reticulation-visible
networks coincide. Thus, in order to use the approach, we need to understand
the class of (not necessarily binary) tree-child networks, which appears in
the first sum of (3). In particular, we need to know which of these networks
contribute to the main term of the asymptotics.

For this, it would be good to have a better understanding of structural
properties of this class. Binary tree-child networks have been counted in
[10]. However, generalizing these results to the non-binary case is non-
trivial. For example, in [3, 4], we considered $d$-combining tree-child
networks which are tree-child networks with all internal vertices either
bifurcating tree vertices or reticulation vertices with one child and exactly
$d$ parents. Extending these results to general tree-child networks seems to
be a major challenge. Such a generalization (and the structural knowledge it
would entail) could, however, be helpful, if one wants to use (3) to find the
asymptotics of ${\rm RV}_{\ell}$.

## Acknowledgment

Both authors acknowledge partially support by NSTC (National Science and
Technology Council, Taiwan) under the research grant
NSTC-111-2115-M-004-002-MY2.

## References

  * [1] M. Bordewich and C. Semple (2016). Reticulation-visible networks, Adv. in Appl. Math., 78, 114–141.
  * [2] G. Cardona and L. Zhang (2020). Counting and enumerating tree-child networks and their subclasses, J. Comput. System Sci., 114, 84–104.
  * [3] Y.-S. Chang, M. Fuchs, H. Liu, M. Wallner, G.-R. Yu (2022). Enumeration of $d$-combining tree-child networks, LIPICS, Proceedings of the 33rd Meeting on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, 225, Paper 5.
  * [4] Y.-S. Chang, M. Fuchs, H. Liu, M. Wallner, G.-R. Yu (2022). Enumerative and distributional results for $d$-combining tree-child networks, arXiv:2209.03850.
  * [5] Y.-S. Chang, M. Fuchs, G.-R. Yu, Galled tree-child networks, in preparation.
  * [6] P. Flajolet and R. Sedgewick. Analytic Combinatorics, 1st edition, Cambridge University Press, Cambridge, 2009.
  * [7] M. Fuchs, B. Gittenberger, M. Mansouri (2019). Counting phylogenetic networks with few reticulation vertices: tree-child and normal networks, Australas. J. Combin., 73:2, 385–423.
  * [8] M. Fuchs, B. Gittenberger, M. Mansouri (2021). Counting phylogenetic networks with few reticulation vertices: exact enumeration and corrections, Australas. J. Combin., 82:2, 257–282.
  * [9] M. Fuchs, E.-Y. Huang, G.-R. Yu (2022). Counting phylogenetic networks with few reticulation vertices: a second approach, Discrete Appl. Math., 320, 140–149.
  * [10] M. Fuchs, G.-R. Yu, L. Zhang (2021). On the asymptotic growth of the number of tree-child networks, European J. Combin., 93, 103278, 20pp.
  * [11] M. Fuchs, G.-R. Yu, L. Zhang (2022). Asymptotic enumeration and distributional properties of galled networks, J. Comb. Theory Ser. A., 189, 105599, 28 pages.
  * [12] P. Gambette, A. D. M. Gunawan, A. Labarre, S. Vialette, L. Zhang (2015). Locating a tree in a phylogenetic network in quadratic time, In Proc. of the 19th Int’l Conf. Res. in Comput. Mol. Biol. (RECOMB), 96–107, Warsaw, Poland.
  * [13] A. D. M. Gunawan, B. DasGupta, L. Zhang (2017). A decomposition theorem and two algorithms for reticulation-visible networks, Inf. Comput., 252, 161–175.
  * [14] A. D. M. Gunawan, J. Rathin, L. Zhang (2020). Counting and enumerating galled networks, Discrete Appl. Math., 283, 644–654.
  * [15] A. D. M. Gunawan, H. Yan, L. Zhang (2019). Compression of phylogenetic networks and algorithm for the tree containment problem, J. Comput. Biol., 26:3, 285–294.
  * [16] A. D. M. Gunawan and L. Zhang (2015) Bounding the size of a network defined by visibility property, arXiv:1510.00115.
  * [17] S. Kong, J. C. Pons, L. Kubatko, K. Wicke (2022). Classes of explicit phylogenetic networks and their biological and mathematical significance, J. Math. Biol., 84, Paper: 47.
  * [18] M. Mansouri (2022). Counting general phylogenetic networks, Australas. J. Combin., 83, 40–86.
  * [19] S. J. Willson (2010). Properties of normal phylogenetic networks, Bull. Math. Biol., 72:2, 340–358.
  * [20] L. Zhang (2019). Generating normal networks via leaf insertion and nearest neighbor interchange, BMC Bioinformatics, 20, Article 642.

## Appendix

The main purpose of this appendix is to prove Theorem 5.

###### Proof of Theorem 5.

We use the component graph method from Section 2. Since the component graph of
a reticulation-visible network is a tree-child network with all vertices of
indegree at most $2$ and no reticulation vertex just followed by one tree
vertex, we first fix such a tree-child network $\tilde{C}$ (without arrows on
the edges).

The maximal number of reticulation vertices a network decompressed from
$\tilde{C}$ can have is obtained by placing arrows on all pendant edges except
the ones directly below reticulation vertices. Let $r(\tilde{C})$ be the
number of these edges plus the number of reticulation vertices of $\tilde{C}$
plus the number of internal vertices with exactly one incoming edge. Then, our
goal is to find those ${\tilde{C}}$ which maximize $r(\tilde{C})$. Note that
$r(\tilde{C})$ remains invariant if we replace vertices with indegree $2$ and
outdegree at least $2$ by a reticulation vertex followed by a tree vertex and
do not count the additional created edge. This is the set of $\tilde{C}$, we
will consider in the sequel. (Thus, for the decompressing procedure, we first
have to merge reticulation vertices followed by just one tree vertices, if
there are any such vertices.)

We start with the following claim

Claim 1: $r(\tilde{C})$ is maximized only for binary tree-child networks
$\tilde{C}$.

Assume that $\tilde{C}$ has at least one vertex, say $v$, with outdegree $\geq
3$. Replacing $v$ with a vertex of outdegree $2$ and attaching to it one of
the children of $v$ and a vertex whose children are the remaining children of
$v$ clearly gives a new tree-child network with $r(\tilde{C})$ increased by
$1$ (since we have created a new internal vertex with indegree $1$). By
iterating this procedure, we end up with a tree-child network which is binary.
This proves our claim.

21 $N$

12 $C(N)$

Figure 6: Left: The smallest example of a maximal reticulated reticulation-
visible network with $2$ leaves (with the tree-components highlighted). Right:
The component graph of the network; note that it is a maximal reticulated
binary tree-child network.

Claim 2: The maximum of $r(\tilde{C})$ over the set of all binary tree-child
networks is $3\ell-3$ and this bound is achieved if and only if $\tilde{C}$ is
a maximal reticulated binary tree-child network.

Note that in any binary tree-child network, we have $\ell+k=t+2$, where $t$ is
the number of tree vertices; see Section 1 in [10]. Thus,

$r(\tilde{C})=\ell+k+t-k=2\ell+k-2.$

Since $k\leq\ell-1$ (see Section 1) with this bound achieved exactly by the
maximal reticulated tree-child networks, the claim follows.

Overall, we have proved so far that the maximal number of reticulation
vertices of a reticulation-visible network is $3\ell-3$ and this bound is
achieved if and only if the component graph of the network is a maximal
reticulated binary tree-child network. Finally, the tree vertices of these
networks have one child which is a reticulation vertex and one child which is
not; see Lemma 1 in [10]. Thus, they are replaced by a one-component network
which has $2$ leaves exactly one of which is below a reticulation vertex.
However, the number of these one-component networks is $M_{2,1}=1$.
Consequently, the decompression of every maximal reticulated binary tree-child
network gives exactly one reticulation-visible network; see Figure 6 for the
smallest example (which was already given in [1]) and Figure 7 for a larger
example. This proves our result.

4312 $N$

4321 $C(N)$

Figure 7: A maximal reticulated reticulation-visible network with $4$ leaves
and $9$ reticulation vertices. The component graph is a maximal reticulated
binary tree-child network. Note that $N$ is the only network obtained by
decompressing $C(N)$.


{}

$missing\text{\,}\mathrm{missing}$

# Wetting and Strain Engineering of 2D Materials on Nanopatterned Substrates

Davoud Adinehloo  Department of Electrical Engineering, University at Buffalo,
Buffalo, NY 14228, USA  Joshua R. Hendrickson  Sensors Directorate, Air Force
Research Laboratory, Wright-Patterson AFB, Ohio 45433, USA  Vasili Perebeinos
<EMAIL_ADDRESS>Department of Electrical Engineering, University at
Buffalo, Buffalo, NY 14228, USA

###### Abstract

The fascinating realm of strain engineering and wetting transitions in two-
dimensional (2D) materials takes place when placed on a two-dimensional array
of nanopillars or one-dimensional rectangular grated substrates. Our
investigation encompasses a diverse set of atomically thin 2D materials,
including transition metal dichalcogenides, hexagonal boron nitride, and
graphene, with a keen focus on the impact of van der Waals adhesion energies
to the substrate on the wetting/dewetting behavior on nanopatterned
substrates. We find a critical aspect ratio of the nanopillar or grating
heights to the period of the pattern when the wetting/dewetting transition
occurs. Furthermore, energy hysteresis analysis reveals dynamic detachment and
re-engagement events during height adjustments, shedding light on energy
barriers of 2D monolayer transferred on patterned substrates. Our findings
offer avenues for strain engineering in 2D materials, leading to promising
prospects for future technological applications.

##  I Introduction

Two-dimensional (2D) materials have garnered immense attention due to their
exceptional electronic Wang  _et al._ (2018, 2012); Avouris  _et al._ (2007),
optical Ma  _et al._ (2021); You  _et al._ (2019), and mechanical Akinwande
_et al._ (2017); Androulidakis  _et al._ (2018) properties. These atomically
thin materials exhibit remarkable promise for various applications, from high-
performance electronics to advanced photonics Lemme  _et al._ (2022);
Ponnusamy  _et al._ (2022). Particularly fascinating is their ability to
function as single-photon emitters (SPEs), a role that holds great promise in
the field of quantum information science and technology Aharonovich  _et al._
(2016); Koenderink  _et al._ (2015); Eisaman  _et al._ (2011).

In the realm of quantum technology, the quest for SPEs has been ongoing, and
2D materials, such as transition metal dichalcogenides (TMD) and hexagonal
boron nitride (hBN), have emerged as promising candidates to fulfill this role
Abramov  _et al._ (2023); Tonndorf  _et al._ (2015); He  _et al._ (2015);
Mendelson  _et al._ (2021). Localization of excitons within 2D materials,
induced by strain or defects, has been identified as the primary source of
such photon emission Abramov  _et al._ (2023); Mendelson  _et al._ (2021);
Montblanch  _et al._ (2021); Kim  _et al._ (2022); Linhart  _et al._ (2019).
Several articles have reported local strain-induced SPEs, providing valuable
insight into their behavior Branny  _et al._ (2017); Liu  _et al._ (2017);
Palacios-Berraquero  _et al._ (2017).

In fact, it is of paramount importance to highlight that the intricate
microscopic mechanisms that govern the formation of strain-induced SPEs in 2D
materials remain a topic of ongoing investigation Azzam  _et al._ (2021).
Further elucidation of these mechanisms is vital to advance our understanding
of SPEs in 2D materials, with direct implications for the development of
reliable and efficient SPEs integrated into quantum technologies An  _et al._
(2022); Tian  _et al._ (2016); Carter  _et al._ (2013); Palacios-Berraquero
_et al._ (2017). The mechanical properties of 2D materials have been
extensively explored Bertolazzi  _et al._ (2011). These properties play an
important role in shaping the electronic and optoelectronic performance of 2D
materials Mahmud  _et al._ (2023); Lei  _et al._ (2022); Touski and Ghobadi
(2021); Adib  _et al._ (2023); Hasani  _et al._ (2023). The adhesion energies
of 2D materials to substrates, for example, have been reported in various
materials: 6 to 28 meV/Å for graphene on silicon oxide Nguyen  _et al._
(2011); Boddeti  _et al._ (2013, 2013); Na  _et al._ (2014); Koenig  _et al._
(2011); Akinwande  _et al._ (2017), 9.4 meV/Å for graphene on silicon Dai  _et
al._ (2020); Zong  _et al._ (2010), 1.12 meV/Å for MoS2 on PMDS Brennan  _et
al._ (2015), 6.3 meV/Å for MoS2 on Al2O3 Sanchez  _et al._ (2018), and 5 meV/Å
for MoS2 on SiO2 Sanchez  _et al._ (2018). Studies on the folding behavior of
2D materials Zhao  _et al._ (2015), and investigations of heterostructures
involving 2D materials Yu  _et al._ (2021), provide valuable information on
their mechanical characteristics. Furthermore, specific structural
arrangements have been examined, such as a graphene membrane stacked on
triangular nanopillars Milovanović  _et al._ (2019), and bubble structures
Blundo  _et al._ (2022, 2021); Chen  _et al._ (2022); Lloyd  _et al._ (2017);
Gastaldo  _et al._ (2023). However, it is essential to note that while several
studies have addressed the mechanical behavior of 2D materials, there is still
a gap in our understanding regarding factors such as geometrical
configuration, interfacial adhesion energy, and the period of the patterned
structure, which warrants further investigation.

Figure 1:  Schematic representations of two structural configurations: (a) a
two-dimensional array of rounded nanopillars and (b) a rectangular one-
dimensional substrate grating. Both nanopatterned substrates are coated with a
monolayer of 2D material. (c) and (d) provide the vertical cross-sectional
view of the 2D material when placed on nanopillars and rectangular grating,
respectively. The key parameters include the period $2a$ of the patterned
substrate, $h$ for the height of the nanostructure, $2r$ representing the
diameter of the nanopillar or the width of the grating, and $h_{a}$ denoting
the height of the 2D material at the edge of the unit cell at $x=a$.

In this study, we offer a significant step towards advancing our ability to
precisely manipulate local strain in 2D materials by gaining a profound
understanding of their mechanical behavior when interfaced with various
substrates. The first configuration that we consider involves an intricate
examination of the geometric variations of 2D materials, incorporating a two-
dimensional array of hard-wall rounded nanopillars (as depicted in Fig. 1(a)).
The intricate surface profile of each pillar, characterized by a half-
semicircular structure with height equal to the radius, that is, $h=r$ (as
depicted in Fig.1(c)), is expertly captured. The second configuration features
a rectangular one-dimensional substrate grating overlaid with a 2D material
filling in the formed trenches, as visualized in Fig. 1(b). Remarkably, the
inherent complexity of these configurations is elegantly distilled into a
concise 1D formulation, as illustrated in both Fig. 1(c) and Fig. 1(d).
Through this research, we provide a deeper understanding of the mechanical
response of 2D materials, opening up new avenues for precision engineering
applications.

##  II Theoretical Framework

In this investigation, we present a rigorous theoretical framework to quantify
the total energy per unit cell area of the patterned structure. This energy is
expressed as follows:

$E=E_{\text{strain}}+E_{\text{bending}}+E_{\text{adhesion}}.$ (1)

where each component $E_{\text{strain}}$, $E_{\text{bending}}$, and
$E_{\text{adhesion}}$, represents a distinct contribution originating from
essential mechanical phenomena: strain, bending, and van der Waals adhesion
energies, respectively. Detailed expressions for these distinct components can
be found in the Appendices A.1 and A.2, corresponding to two-dimensional
nanopillar array and one-dimensional grating configurations, respectively.

To assess the strain experienced by the 2D material, we utilize the following
equation:

$\varepsilon(x)=\sqrt{1+y^{\prime}(x)^{2}}-1.$ (2)

We assumed here that the area of 2D material equals the unit cell area of
$2a\times 2a$ and no sliding is allowed, such that any tilt of 2D material
results in the local strain given by Eq. (2).

##  III Results and discussion

Figure 2:  Engineering MoSe2 structure on two-dimensional nanopillar array and
one-dimensional grating configurations. (a) Demonstrates the customized
behavior of MoSe2 on nanopillar array, while (b) illustrates the controlled
modulation of the MoSe2 structure on grated substrate with varying interfacial
adhesion energy ($\gamma$). (c) Unveils the intricate strain distribution
within the MoSe2 structures on nanopillars, and (d) presents the comprehensive
strain distribution in the MoSe2 structures in the grated substrates over a
range of $\gamma$ levels. These figures collectively highlight the profound
impact of the interfacial adhesion energy ($\gamma$) on MoSe2 positioning and
the distribution of strain. A constant value $a$ of 50 nm is maintained in all
configurations to ensure a consistent basis for comparison. The gray area in
(a) and (b) represents the nanostructured substrate.

Figure 2 illustrates the mechanical response of MoSe2, which serves as a
representative example of 2D materials when strategically placed on a two-
dimensional array of nanopillars or on a rectangular one-dimensional substrate
grating. Since the adhesion energy can be engineered by choosing a 2D material
and substrate, we present our results for an experimentally accessible range
of adhesion energies $\gamma$.

In Figure 2(a), we observe the precise tailoring of MoSe2 behavior when placed
on a nanopillar array. This figure provides visual evidence of van der Waals
interactions that govern the material’s positioning atop the nanopillars.
Manipulation of the MoSe2 geometry on one-dimensional substrate grating with
varying values $\gamma$ is demonstrated in Figure 2(b). This highlights the
versatility of MoSe2 in adapting to different interfacial adhesion energies.

Figure 2(c) provides an in-depth exploration of the strain distribution within
the MoSe2 structures on nanopillars. It becomes evident that the strained area
of the 2D material can be manipulated by varying the interfacial adhesion
energy. As $\gamma$ increases or decreases, the extent of the strain within
the material adjusts accordingly, leading to a controlled modulation of the
strained area. This tunability is of paramount importance in tailoring the
mechanical response of 2D materials for specific applications, particularly in
SPEs applications.

In Figure 2(d), we extend a similar analysis to MoSe2 structures within
grating configurations, providing a detailed view of its response to varying
$\gamma$ values. Interestingly, a distinct behavior is observed in grated
structures: the strain in the strained area remains relatively constant until
the 2D material fully contacts the substrate in the trenches. This phenomenon
suggests that grating configurations exhibit a unique mechanical response
characterized by a sudden change in strain behavior upon contact, distinct
from the gradual modulation observed in nanopillar arrays.

Figure 3:  The evolution of the critical distance ratio ($r_{c}$) relative to
the heights of the nanopillars and the depths of the grating in response to
the variations $\gamma$ between different 2D materials and substrates is
presented in (a) and (d), respectively. The maximum strain within the 2D
material is analyzed in (b) for nanopillar array configurations and (e) for
grating configurations as a function of the interfacial adhesion energy. The
average strain within the 2D material is illustrated in (c) for the nanopillar
array and in (f) for grating scenarios.

To dig deeper into the response of 2D materials placed in nanopillar arrays
and patterned substrates, we introduce the concept of a critical distance,
denoted $r_{c}$. This critical distance represents the point at which the
strain levels approach near-zero values. Figure 3 presents the ratio of the
critical distance $r_{c}$ to $h$, together with the maximum and average strain
values as functions of $\gamma$.

Figs. 3(a) and 3(d) depict the ratio of $r_{c}/h$ versus interfacial adhesion
energy for the case of the nanopillar array and the patterned substrate,
respectively. In particular, as $\gamma$ increases, this ratio decreases
consistently. Importantly, this trend remains consistent over varying periods.
Furthermore, the ratio exhibits variations between different materials,
primarily due to differences in their Young modulus and bending stiffness, as
detailed in Table 1.

The evolution of maximum strain as a function of $\gamma$ is depicted in Figs.
3(b) and 3(e) for nanopillar array and the patterned substrate, respectively.
It becomes evident that as $\gamma$ increases, the maximum strain exhibits a
corresponding increase, due to the changes in the shape of the 2D material
induced by stronger adhesion between its layers. Additionally, it should be
noted that the maximum strain for 2D materials on the nanopillar arrays is
approximately twice that observed for 2D materials atop one-dimensional
patterned substrates.

Since exciton emission can occur from different locations in 2D material, we
introduce the concept of average strain as follows:

$\varepsilon_{ave}=\dfrac{1}{r_{c}}\int_{0}^{r_{c}}\varepsilon(x)dx,$ (3)

and illustrate it as a function of $\gamma$ for different materials in Figs.
3(c) and 3(f) for nanopillar array and rectangular grating of the substrate,
respectively. Evidently, the average strain exhibits an upward trajectory with
increasing interfacial adhesion energy. However, it should be noted that, in
contrast to the differences in maximum strain, the average strain values for
nanopillar arrays and grating configurations are observed to be in close
proximity to each other.

Figure 4:  Energy hysteresis and analysis of the activation energy. (a) and
(b) show the energy hysteresis for the nanopillar array configuration and the
grated substrate, respectively, as a function of $h_{a}$ for different lengths
of the period. (c) and (d) show the activated energy for the nanopillar array
configuration and the grated substrate configuration, respectively, as a
function of periods across diverse materials. The adhesion energy is set to 1
meV/$$\mathrm{\text{\r{A}}}$^{2}$ in both cases. These comprehensive analyses
provide deep insight into the energy behavior of these structures across
various parameters.

Figure 4 presents a detailed analysis of the energy variations within the
sample structures. Specifically, Figs. 4 (a) and 4 (b) provide information on
the energy hysteresis observed with respect to changes in $h_{a}$. In
particular, as the parameter $h_{a}$ increases, an intriguing phenomenon
unfolds: The 2D material gradually disengages from the substrate, leading to a
gradual increase in overall energy. However, a critical threshold of $h_{a}$
is reached, beyond which a remarkable event occurs, the complete detachment of
the 2D material from the substrate. This detachment event is characterized by
a sharp and pronounced drop in total energy. Beyond this critical threshold,
further increases in $h_{a}$ no longer significantly impact the energy of the
system as it stabilizes in a detached state. Conversely, when reducing
$h_{a}$, the energy landscape exhibits a distinct behavior. Initially, as
$h_{a}$ decreases, the energy remains relatively constant until it reaches a
lower value. Subsequently, a departure from this equilibrium occurs, resulting
in an increase in energy. Ultimately, the energy reaches the activation
energy, denoted $E_{a}$, signifying the point at which the 2D material re-
establishes contact with the substrate. This reengagement prompts a swift and
substantial drop in energy as the structure strives to achieve its optimized
configuration.

Turning our attention to Figs. 4(c) and 4(d), these plots provide a detailed
analysis of the activation energy as a function of the period in a range of
materials. A clear and consistent trend emerges: as the period increases, the
activation energy exhibits a consistent decrease. This observation underscores
the intriguing relationship between the period and the energy required for
structural relaxation. Specifically, longer periods are associated with a
reduced demand for energy during the relaxation process, a phenomenon of
significance in the study of material behavior.

##  IV Conclusion

In summary, our advanced theoretical framework provides a comprehensive
understanding of the mechanics of 2D material membranes when transferred to
different patterned substrates, such as nanopillar array and grating
structures. It elucidates the critical role of the interfacial adhesion energy
($\gamma$) in the governing behavior of these materials. As $\gamma$
increases, significant alterations are observed in the strain, critical
period, and energy profiles, highlighting the sensitivity of 2D materials to
this parameter and offering the prospect of precise engineering for customized
nanoscale applications. A notable outcome of this study is the concept of a
critical aspect ratio of the height of the nanostructure to the period
($r_{c}$), which emerges as a key metric that characterizes the response of
the system. $r_{c}$ varies with both $\gamma$ and the specific material,
providing valuable information on how 2D materials adapt and interact with
different substrate environments.

The analysis of energy hysteresis reveals intriguing detachment and
reengagement events as the height of the 2D material on the substrate changes.
These events are reflected in sharp fluctuations in energy profiles,
emphasizing the dynamic nature of these systems and the energy barriers
associated with structural rearrangements. Furthermore, the investigation of
activation energy demonstrates that it decreases with a larger period,
shedding light on the role of the pattern period in shaping material behavior.
This advanced framework not only enhances our understanding of 2D material
mechanics but also presents opportunities for precision engineering, paving
the way for tailored nanoscale materials and devices with specific mechanical
properties, thus contributing to the evolving field of 2D materials and their
potential technological applications.

##  Appendix A Energy Contributions

The geometric profile of the 2D material overlaid on nanopillar arrays and
grating structures can be described through a Fourier series, as defined by:

$y(x)=A_{0}+\sum_{n=1}^{N}A_{n}\cos\left(\frac{n\pi x}{a}\right).$ (4)

Upon enforcing boundary conditions, this representation simplifies to:

$y(x)=h_{a}+\sum_{n=1}^{N}A_{n}[\cos(\dfrac{n\pi x}{a})-\cos(n\pi)].$ (5)

Here, $h_{a}$ denotes the height of the 2D material at $x=a$ (see Fig. 1),
while $2a$ represents the period of the structure, and $A_{n}$ signifies the
Fourier coefficient. In this section, we discuss the components that
contribute to the total energy of the 2D profile, encompassing the stresses,
bending, and van der Waals adhesion energies.

###  A.1 2D Materials on Nanopillar Arrays

To elucidate the underlying energy expressions that govern this configuration,
we present the following equations.

The quantification of strain energy is expressed as:

$E_{\text{strain}}=\pi Y\int_{0}^{a}x\varepsilon(x)^{2}dx,$ (6)

where $Y$ represents the Young’s modulus.

Table 1:  Mechanical Characteristics of Various Materials, including Young’s Modulus ($Y$), Bending Stiffness ($\chi$), and Monolayer Thickness ($d$) Yang  _et al._ (2017); Lee  _et al._ (2008). Materials | Y (GPa) |  $\chi$ (eV) |  $d$ (nm)
---|---|---|---
MoSe2 | 177 | 9.96 | 0.700
Graphene | 1000 | 1.40 | 0.335
hBN | 865 | 0.86 | 0.334

The energy associated with bending is articulated as:

$E_{\text{bending}}=\pi\chi\int_{0}^{a}x\frac{y^{\prime\prime}(x)^{2}}{(1+y^{\prime}(x)^{2})^{3}}dx,$
(7)

with $\chi$ denoting the bending stiffness. The values for $Y$ and $\chi$ in
various materials are provided in Table 1.

To evaluate adhesion energy, we employ the Morse potential Morse (1929)
between layers:

$\displaystyle E_{\text{adhesion}}=$ $\displaystyle\pi
4a^{2}\gamma+(4-\pi)a^{2}U_{2D}(y(a))$ (8)
$\displaystyle+2\pi\int_{0}^{a}x~{}U_{2D}(y(x))dx,$

where $\gamma$ represents the interfacial adhesion energy between the 2D
material and the substrate. The Morse potential, $U_{2D}(y)$, is given by:

$U_{2D}(y)=\gamma\left(e^{-2\sqrt{5}((y/h_{0})^{2}-1)}-2e^{-\sqrt{5}((y/h_{0})^{2}-1)}\right).$
(9)

Here, $h_{0}=0.34~{}nm$ stands for the equilibrium distance between layers.
Within these equations, $y^{\prime}(x)$ and $y^{\prime\prime}(x)$ represent
the first and second derivatives of the shape of the 2D material,
respectively. This comprehensive ensemble of energy components collectively
serves to elucidate the intricate mechanical interactions between 2D materials
and nanopillar arrays.

###  A.2 2D Material on rectangular grating substrate

Another intriguing configuration involves positioning 2D materials over a
grating structure. The fundamental energy expressions governing this
configuration are described in detail below.

The strain energy can be obtained by:

$E_{\text{strain}}=Ya\int_{0}^{a}\varepsilon(x)^{2}dx.$ (10)

To calculate the bending energy, we employ:

$E_{\text{bending}}=a\chi\int_{0}^{a}\frac{h^{\prime\prime}(x)^{2}}{(1+h^{\prime}(x)^{2})^{3}}dx.$
(11)

Finally, we calculate the adhesion energy using:

$\displaystyle E_{\text{adhesion}}=$ $\displaystyle
4a^{2}\gamma+4a\int_{0}^{a}U_{2D}(y(x))dx$ (12)
$\displaystyle+4a\int_{0}^{a}dx\int_{-r}^{r}\bigl{(}U_{1D}(x-x^{\prime},y(x)-r)$
$\displaystyle-U_{1D}(x-x^{\prime},y(x))\bigr{)}dx^{\prime}$
$\displaystyle+4a\int_{0}^{a}dx\int_{0}^{r}U_{1D}(x-r,y(x)-y^{\prime})dy^{\prime},$

where $U_{1D}(x,y)$ is described as:

$\displaystyle U_{1D}(x,y)=$ $\displaystyle\gamma\frac{2\times
5^{1/4}}{\sqrt{\pi}h_{0}}$ (13)
$\displaystyle\left(\frac{1}{\sqrt{2}}e^{-2\sqrt{5}((x^{2}+y^{2})/h_{0}^{2}-1)}-e^{-\sqrt{5}((x^{2}+y^{2})/h_{0}^{2}-1)}\right).$

###### Acknowledgements.

We gratefully acknowledge the support of the Air Force Office of Scientific
Research under award number FA9550-22-1-0312 and the computational facilities
of the Center for Computational Research at the University of Buffalo
(http://hdl.handle.net/10477/79221). J.R.H. acknowledges support from the Air
Force Office of Scientific Research (Program Manager, Dr. Gernot Pomrenke)
under award number FA9550-20RYCOR059.

## References

  * Wang  _et al._ (2018) G. Wang, A. Chernikov, M. M. Glazov, T. F. Heinz, X. Marie, T. Amand, and B. Urbaszek, Rev. Mod. Phys. 90, 021001 (2018).
  * Wang  _et al._ (2012) Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman, and M. S. Strano, Nature nanotechnology 7, 699 (2012).
  * Avouris  _et al._ (2007) P. Avouris, Z. Chen, and V. Perebeinos, Nature nanotechnology 2, 605 (2007).
  * Ma  _et al._ (2021) Q. Ma, G. Ren, K. Xu, and J. Z. Ou, Advanced Optical Materials 9, 2001313 (2021).
  * You  _et al._ (2019) J. You, S. Bongu, Q. Bao, and N. Panoiu, Nanophotonics 8, 63 (2019).
  * Akinwande  _et al._ (2017) D. Akinwande, C. J. Brennan, J. S. Bunch, P. Egberts, J. R. Felts, H. Gao, R. Huang, J.-S. Kim, T. Li, Y. Li, K. M. Liechti, N. Lu, H. S. Park, E. J. Reed, P. Wang, B. I. Yakobson, T. Zhang, Y.-W. Zhang, Y. Zhou, and Y. Zhu, Extreme Mechanics Letters 13, 42 (2017).
  * Androulidakis  _et al._ (2018) C. Androulidakis, K. Zhang, M. Robertson, and S. Tawfick, 2D Materials 5, 032005 (2018).
  * Lemme  _et al._ (2022) M. C. Lemme, D. Akinwande, C. Huyghebaert, and C. Stampfer, Nature communications 13, 1392 (2022).
  * Ponnusamy  _et al._ (2022) M. Ponnusamy, K. Ramya, V. Sivasankaran, H. Farmani, and A. Farmani, Journal of Materials Research , 1 (2022).
  * Aharonovich  _et al._ (2016) I. Aharonovich, D. Englund, and M. Toth, Nature photonics 10, 631 (2016).
  * Koenderink  _et al._ (2015) A. F. Koenderink, A. Alù, and A. Polman, Science 348, 516 (2015).
  * Eisaman  _et al._ (2011) M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, Review of Scientific Instruments 82, 071101 (2011).
  * Abramov  _et al._ (2023) A. N. Abramov, I. Y. Chestnov, E. S. Alimova, T. Ivanova, I. S. Mukhin, D. N. Krizhanovskii, I. A. Shelykh, I. V. Iorsh, and V. Kravtsov, Nature Communications 14, 5737 (2023).
  * Tonndorf  _et al._ (2015) P. Tonndorf, R. Schmidt, R. Schneider, J. Kern, M. Buscema, G. A. Steele, A. Castellanos-Gomez, H. S. J. van der Zant, S. M. de Vasconcellos, and R. Bratschitsch, Optica 2, 347 (2015).
  * He  _et al._ (2015) Y.-M. He, G. Clark, J. R. Schaibley, Y. He, M.-C. Chen, Y.-J. Wei, X. Ding, Q. Zhang, W. Yao, X. Xu, _et al._ , Nature nanotechnology 10, 497 (2015).
  * Mendelson  _et al._ (2021) N. Mendelson, D. Chugh, J. R. Reimers, T. S. Cheng, A. Gottscholl, H. Long, C. J. Mellor, A. Zettl, V. Dyakonov, P. H. Beton, _et al._ , Nature materials 20, 321 (2021).
  * Montblanch  _et al._ (2021) A. R.-P. Montblanch, D. M. Kara, I. Paradisanos, C. M. Purser, M. S. Feuer, E. M. Alexeev, L. Stefan, Y. Qin, M. Blei, G. Wang, _et al._ , Communications Physics 4, 119 (2021).
  * Kim  _et al._ (2022) G. Kim, H. M. Kim, P. Kumar, M. Rahaman, C. E. Stevens, J. Jeon, K. Jo, K.-H. Kim, N. Trainor, H. Zhu, B.-H. Sohn, E. A. Stach, J. R. Hendrickson, N. R. Glavin, J. Suh, J. M. Redwing, and D. Jariwala, ACS Nano 16, 9651 (2022).
  * Linhart  _et al._ (2019) L. Linhart, M. Paur, V. Smejkal, J. Burgdörfer, T. Mueller, and F. Libisch, Phys. Rev. Lett. 123, 146401 (2019).
  * Branny  _et al._ (2017) A. Branny, S. Kumar, R. Proux, and B. D. Gerardot, Nature communications 8, 15053 (2017).
  * Liu  _et al._ (2017) Y. Liu, W. Huang, T. Gong, Y. Su, H. Zhang, Y. He, Z. Liu, and B. Yu, Nanoscale 9, 17459 (2017).
  * Palacios-Berraquero  _et al._ (2017) C. Palacios-Berraquero, D. M. Kara, A. R.-P. Montblanch, M. Barbone, P. Latawiec, D. Yoon, A. K. Ott, M. Loncar, A. C. Ferrari, and M. Atatüre, Nature communications 8, 15093 (2017).
  * Azzam  _et al._ (2021) S. I. Azzam, K. Parto, and G. Moody, Applied Physics Letters 118, 240502 (2021).
  * An  _et al._ (2022) J. An, X. Zhao, Y. Zhang, M. Liu, J. Yuan, X. Sun, Z. Zhang, B. Wang, S. Li, and D. Li, Advanced Functional Materials 32, 2110119 (2022).
  * Tian  _et al._ (2016) H. Tian, M. L. Chin, S. Najmaei, Q. Guo, F. Xia, H. Wang, and M. Dubey, Nano Research 9, 1543 (2016).
  * Carter  _et al._ (2013) S. G. Carter, T. M. Sweeney, M. Kim, C. S. Kim, D. Solenov, S. E. Economou, T. L. Reinecke, L. Yang, A. S. Bracker, and D. Gammon, Nature Photonics 7, 329 (2013).
  * Bertolazzi  _et al._ (2011) S. Bertolazzi, J. Brivio, and A. Kis, ACS Nano 5, 9703 (2011).
  * Mahmud  _et al._ (2023) M. T. Mahmud, D. Zhai, and N. Sandler, Nano Letters 23, 7725 (2023).
  * Lei  _et al._ (2022) Y. Lei, T. Zhang, Y.-C. Lin, T. Granzier-Nakajima, G. Bepete, D. A. Kowalczyk, Z. Lin, D. Zhou, T. F. Schranghamer, A. Dodda, A. Sebastian, Y. Chen, Y. Liu, G. Pourtois, T. J. Kempa, B. Schuler, M. T. Edmonds, S. Y. Quek, U. Wurstbauer, S. M. Wu, N. R. Glavin, S. Das, S. P. Dash, J. M. Redwing, J. A. Robinson, and M. Terrones, ACS Nanoscience Au 2, 450 (2022).
  * Touski and Ghobadi (2021) S. B. Touski and N. Ghobadi, Journal of Physics D: Applied Physics 54, 485302 (2021).
  * Adib  _et al._ (2023) H. Adib, M. Mazaherifar, S. B. Touski, and S. Mohajerzadeh, IEEE Transactions on Electron Devices , 1 (2023).
  * Hasani  _et al._ (2023) N. Hasani, A. Rajabi-Maram, and S. B. Touski, Journal of Physics and Chemistry of Solids 174, 111131 (2023).
  * Nguyen  _et al._ (2011) T. C. Nguyen, M. Otani, and S. Okada, Phys. Rev. Lett. 106, 106801 (2011).
  * Boddeti  _et al._ (2013) N. G. Boddeti, S. P. Koenig, R. Long, J. Xiao, J. S. Bunch, and M. L. Dunn, Journal of Applied Mechanics 80, 040909 (2013).
  * Na  _et al._ (2014) S. R. Na, J. W. Suk, R. S. Ruoff, R. Huang, and K. M. Liechti, ACS Nano 8, 11234 (2014).
  * Koenig  _et al._ (2011) S. P. Koenig, N. G. Boddeti, M. L. Dunn, and J. S. Bunch, Nature nanotechnology 6, 543 (2011).
  * Dai  _et al._ (2020) Z. Dai, N. Lu, K. M. Liechti, and R. Huang, Current Opinion in Solid State and Materials Science 24, 100837 (2020).
  * Zong  _et al._ (2010) Z. Zong, C.-L. Chen, M. R. Dokmeci, and K.-t. Wan, Journal of Applied Physics 107, 026104 (2010).
  * Brennan  _et al._ (2015) C. J. Brennan, J. Nguyen, E. T. Yu, and N. Lu, Advanced Materials Interfaces 2, 1500176 (2015).
  * Sanchez  _et al._ (2018) D. A. Sanchez, Z. Dai, P. Wang, A. Cantu-Chavez, C. J. Brennan, R. Huang, and N. Lu, Proceedings of the National Academy of Sciences 115, 7884 (2018).
  * Zhao  _et al._ (2015) J. Zhao, Q. Deng, T. H. Ly, G. H. Han, G. Sandeep, and M. H. Rümmeli, Nature communications 6, 8935 (2015).
  * Yu  _et al._ (2021) J. Yu, E. Han, M. A. Hossain, K. Watanabe, T. Taniguchi, E. Ertekin, A. M. van der Zande, and P. Y. Huang, Advanced Materials 33, 2007269 (2021).
  * Milovanović  _et al._ (2019) S. P. Milovanović, L. Covaci, and F. M. Peeters, Journal of Applied Physics 125, 082534 (2019).
  * Blundo  _et al._ (2022) E. Blundo, A. Surrente, D. Spirito, G. Pettinari, T. Yildirim, C. A. Chavarin, L. Baldassarre, M. Felici, and A. Polimeni, Nano Letters 22, 1525 (2022).
  * Blundo  _et al._ (2021) E. Blundo, T. Yildirim, G. Pettinari, and A. Polimeni, Phys. Rev. Lett. 127, 046101 (2021).
  * Chen  _et al._ (2022) Y. Chen, Y. Wang, W. Shen, M. Wu, B. Li, Q. Zhang, S. Liu, C. Hu, S. Yang, Y. Gao, and C. Jiang, ACS Nano 16, 16271 (2022).
  * Lloyd  _et al._ (2017) D. Lloyd, X. Liu, N. Boddeti, L. Cantley, R. Long, M. L. Dunn, and J. S. Bunch, Nano Letters 17, 5329 (2017).
  * Gastaldo  _et al._ (2023) M. Gastaldo, J. Varillas, Á. Rodríguez, M. Velickỳ, O. Frank, and M. Kalbáč, npj 2D Materials and Applications 7, 71 (2023).
  * Yang  _et al._ (2017) Y. Yang, X. Li, M. Wen, E. Hacopian, W. Chen, Y. Gong, J. Zhang, B. Li, W. Zhou, P. M. Ajayan, Q. Chen, T. Zhu, and J. Lou, Advanced Materials 29, 1604201 (2017).
  * Lee  _et al._ (2008) C. Lee, X. Wei, J. W. Kysar, and J. Hone, Science 321, 385 (2008).
  * Morse (1929) P. M. Morse, Phys. Rev. 34, 57 (1929).