confidential

# Exact Counts of $C_{4}$s in Blow-Up Graphs

S. Y. Chan Deakin University, Geelong, Australia, School of Information
Technology, Faculty of Science Engineering & Built Environment, Australia K.
Morgan††footnotemark:  J.Ugon11footnotemark: 1

###### Abstract

Cycles have many interesting properties and are widely studied in many
disciplines. In some areas, maximising the counts of $k$-cycles are of
particular interest. A natural candidate for the construction method used to
maximise the number of subgraphs $H$ in a graph $G$, is the _blow-up_ method.
Take a graph $G$ on $n$ vertices and a pattern graph $H$ on $k$ vertices, such
that $n\geq k$, the blow-up method involves an iterative process of replacing
vertices in $G$ with a copy of the $k$-vertex graph $H$. In this paper, we
apply the blow-up method on the family of cycles. We then present the exact
counts of cycles of length 4 for using this blow-up method on cycles and
generalised theta graphs.

##  1 Introduction

In graph theory, the family of cycles are considered to have very rich
structures. Cycles have many interesting properties and are of interest in
many disciplines. Some of the many applications of cycles include periodic
scheduling, the identification of weak interdependence in ecosystems and to
determine chemical pathways in chemical networks [1, 9].

In network analysis, cycles are used in modelling and studying communication
systems, improve consensus network performance, to investigate faults and
reliability and also study the topological features of such networks [16]. In
some cases, counting cycles is used as part of network analysis or message-
passing algorithms [8, 11, 15]. Some other interesting problems arise in
relation to counting $k$-cycles in graphs. For example, counting the number of
4-cycles in a tournament [10] and enumerating cycles of a given length [2].

Further, some problems looks into maximising the number of cycles in graphs.
There exists a kidney exchange program (KEP), which involves maximising the
number of (directed) cycles to maximise the expected number of transplants [3,
4, 13].

In the areas of extremal graph theory, a construction method used to maximise
counts of graphs is the _blow-up_ method. This method has been applied in
graphs to study graph spectra [12] and even the maximum induced density of
graphs [6]. This method was also applied as an approach to the Caccetta-
Häggkivst conjecture [5] and Johansson conjecture [7].

Suppose we have graphs $G$ and $H$ on $n$ and $k$ vertices respectively, such
that $n\geq k$. The blow-up method is an iterative process of replacing each
vertex in $G$ with a _copies_ of $H$. If _all_ vertices in $G$ have been
replaced with a copy of $H$, this is also known as a _balanced_ blow-up of
$G$.

In this paper, we are interested in determining the exact number of induced
$C_{4}$s in the nested blow-up graph. We will find the exact counts of
$C_{4}$s in two different blow-up graphs, one in the nested blow-up of
$C_{4}$s and the other in the theta graph $\Theta_{2,2,2}$. We give the
associated formulas with respect to the level of blow-up $N$, which is defined
in the later section.

##  2 Notations and Definitions

In this section, we present some basic definitions and notations that are used
in this paper. All graphs in this paper are simple.

A graph $G=(V,E)$ consists of the (finite) vertex set $V$ and edge set $E$,
which is a subset of all unordered pair of vertices. The _order_ of a graph is
the number of vertices, whereas the _size_ of a graph is the number of edges.
Let $u,v\in V(G)$, we say that $u$ is _adjacent_ to $v$ if there exists an
edge $\\{u,v\\}\in E(G)$. We say that the edge $\\{u,v\\}$ is incident to
vertices $u$ and $v$.

######  Definition 1 (Graph composition).

Let $G$ and $H$ be simple graphs. The composition of graph $G$ and $H$,
denoted $G[H]$, is the graph obtained by replacing each vertex $v\in V(G)$ by
the graph $H_{v}\cong H$ and adding an edge between every vertex of $H_{u}$
and every vertex of $H_{v}$ where $\\{u,v\\}\in E(G).$

######  Definition 2 (Nested blow-up graph).

If $H$ is isomorphic to $G$ and each vertex $v\in V(G)$ is labelled
$v\in\\{0,1,2,\ldots,n-1\\}$, then $G_{0}=G$, $G_{1}=G[G]$,$\ldots$,
$G_{N}=G[\underbrace{G[G[G[\ldots[G]]]]}_{N-1}]=G[G_{N-1}]$, where at each
level $N$, the nested blow-up graph $G_{N}$ can be obtained from $G_{N-1}$ by
replacing each vertex in $G$ by a copy of $G_{N-1}$ and replacing each edge by
all possible edges between adjacent copies of $G_{N-1}$. We refer to each of
such copies $G_{N-1}$ as blobs in our proofs in order to differentiate from
other subgraphs in $G_{N}$ that may be isomorphic to $G_{N-1}$.

$v_{1}$$v_{2}$$K_{2}$$v_{1}$$v_{2}$$v_{1}^{\prime}$$v_{2}^{\prime}$$K_{2}[K_{2}]$
Figure 1: An example of the nested blow-up of $K_{2}$. In $K_{2}[K_{2}]$ there
are two blobs shown in blue and red, but 6 subgraphs that are isomorphic to
$K_{2}$.  Figure 2: Example of a nested blow-up graph of $C_{4}$.

We say that blob $i$ and blob $j$ in $G_{N}$ are adjacent, denoted
$i\circledast j$, if $\\{v_{i},v_{j}\\}\in E(G)$, for $i\in[0,n),j\in[0,n)$.
The vertices of $G_{N}$ are $\\{0,1,\ldots,n|V(G_{N-1})|-1\\}$, where
$|V(G)|=n$. The edges are $\\{\\{u+i\times n^{N},v+i\times
n^{N}\\}:i\in[0,n),\\{u,v\\}\in E(G_{N-1})\\}\cup\\{\\{u+i\times
n^{N},v+i+j\times n^{N}\\}\pmod{|V(G_{N})|}\\}:i\in[0,n),j\in[0,n),u,v\in
V(G),i\circledast j\\}$. There exists copies of subgraphs that are isomorphic
to $G$ within $G_{N-1}$ and also between each blob.

##  3 Exact Counts

In this section, we show the exact counts of $C_{4}$ for the nested blow-up
graph of the graph $G=C_{4}$ and $G=\Theta_{2,2,2}$. Here, we denote $T_{N}$
as the number of induced subgraphs isomorphic to $C_{4}$ and $n_{N}$ as the
number of vertices at level $N$ of the nested blow-up $G_{N}$ respectively
(i.e., $T_{0}=1$ and $n_{0}=4$), where $G_{0}=C_{4}$. We also denote the
number of non-edges of $C_{4}$ as $m_{N}^{c}$, where $m_{0}^{c}=2$. Recall
that a _blob_ is the set of $G_{N-1}$ vertices in $G_{N}$ from the blow-up of
a vertex $v$ in $G=G_{0}$. We show the following lemma:

######  Lemma 1.

The number of non-edges $m^{c}_{N}$ in the nested blow-up graph $G_{N}$ is

$m_{N}^{c}=\binom{4^{N+1}}{2}-4^{N+1}{\sum}_{i=0}^{N}4^{i}=\dfrac{4^{N+1}(4^{N+1}-1)}{6},N\in\mathbb{N}\cup\\{0\\}.$

###### Proof.

We prove the equation from Lemma 1 by induction.
Base case: For $N=0$, the number of non-edges is
$m^{c}_{0}=\binom{4}{2}-4{\sum}^{0}_{i=0}4^{0}=6-4=\dfrac{4^{1}(4^{1}-1)}{6}=2$,
which is precisely the number of non-edges in $C_{4}$.
Assume the induction hypothesis that for a particular $N$, the case $n=N$
holds, that is:

$\displaystyle m^{c}_{N}$
$\displaystyle=\binom{4^{N+1}}{2}-4^{N+1}{\sum}^{N}_{i=0}4^{i}=\dfrac{4^{N+1}(4^{N+1}-1)}{6}.$
The number of non-edges in $G_{N+1}$ is: $\displaystyle m^{c}_{N+1}$
$\displaystyle=\binom{4^{N+2}}{2}-4\times|E(G_{N})|-4\times\text{edges between
pairs of blobs}$ We obtain $|E(G_{N})|$ using $m_{N}^{c}$ such that
$|E(G_{N})|=\binom{4^{N+1}}{2}-m_{N}^{c}$. Thus, $\displaystyle m_{N+1}^{c}$
$\displaystyle=\binom{4^{N+2}}{2}-4\cdot
4^{N+1}{\sum}^{N}_{i=0}4^{i}-4\cdot(4^{N+1})^{2}$
$\displaystyle=\binom{4^{N+2}}{2}-4^{N+2}{\sum}_{i=0}^{N}4^{i}-4^{N+2}4^{N+1}$
$\displaystyle=\binom{4^{N+2}}{2}-4^{N+2}{\sum}_{i=0}^{N+1}4^{i}.$

Since both the base case and inductive step have been proven as true, thus by
mathematical induction $m^{c}_{N}$ holds for all $N$. ∎

Thus, we state the following theorem.

######  Theorem 1.

The nested blow-up graph $G_{N}$, $N\geq 0$, of a $C_{4}$ has precisely
$T_{N}=\frac{8\times 4^{N}}{5670}\times(1280\times 4^{3N}+672\times
4^{2N}+105\times 4^{N}-713)$ induced subgraphs isomorphic to $C_{4}$.

###### Proof.

First, we will show that,

$T_{N}=\begin{cases}4(T_{N-1})+(4^{N})^{4}+4\times
m_{N-1}^{c}\times(4^{N})^{2}+4\times(m_{N-1}^{c})^{2},&n>0\\\
1&n=0.\end{cases}$

Since $G_{0}=C_{4}$, we have $T_{0}=1$. We show that we can obtain $T_{N}$
from $T_{N-1}$ and prove each term from $T_{N}$ respectively.

$B_{1}$$B_{4}$$B_{2}$$B_{3}$ Figure 3: Illustration of $G_{N}$ with four blobs
labelled $B_{1},B_{2},B_{3}$ and $B_{4}$.

$B_{4}$$B_{3}$

$B_{3}$

Figure 4: [Left to right] Illustration of ways to obtain $C_{4}$ from either
2, 3, or all 4 blobs respectively. The selected vertices are shown in red.

We know that $G_{N}$ has four blobs (see Figure 3) isomorphic to $G_{N-1}$.
Each of these blobs has $T_{N-1}$ copies of $C_{4}$, which contributes to
$4T_{N-1}$ induced copies of $C_{4}$. Thus giving the first term $4T_{N-1}$.

For simplicity of the remainder of the proof, we use Figure 4 to show the
multiple ways of obtaining an induced $C_{4}$ from either 2, 3 or all 4 blobs.
Example of how vertices can be chosen are shown in red.

We select a vertex from each of the 4 blobs that are isomorphic to $G_{N-1}$
in $G_{N}$ which results in an induced $C_{4}$. There are $4^{N}$ vertices in
each $G_{N-1}$ which gives $(4^{N})^{4}$ choices. This results in the second
term $(4^{N})^{4}$.

For the third term $4\times m^{c}_{N-1}\times(4^{N})^{2}$, we choose 1 blob
that is isomorphic to $G_{N-1}$. We choose a non-edge from this blob. Then, we
select two adjacent blobs and from each blob we select a vertex. There are
$(4^{N})$ vertices in a blob. As we choose a different vertex from each of the
2 blobs, there are $(4^{N})^{2}$ choices of vertices, as well as $4\times
m^{c}_{n-1}$ non-edges, that can be chosen from the third blob.

Lastly, we choose 2 adjacent blobs and a non-edge from each. There are 4
choices for adjacent blobs in the blow-up of $C_{4}$. Thus, resulting in
$4\times(m^{c}_{N-1})^{2}$ choices.

We now expand and simplify $T_{N}$ to find the recurrence relation.

Expanding the first few terms, we obtain:

$\displaystyle T_{0}={}$ $\displaystyle 1$ $\displaystyle T_{1}={}$
$\displaystyle 4(T_{0})+4^{4}+4\times m_{0}^{c}\times
4^{2}+4\times(m_{0}^{c})^{2}={}4+4^{4}+4^{3}\times
m_{0}^{c}+4\times(m_{0}^{c})^{2}$ $\displaystyle={}$ $\displaystyle
4^{1}\times\sum_{i=0}^{1}4^{3i}+{\sum}_{i=1}^{1}(4^{1+i+1}\times
m_{i-1}^{c})+{\sum}_{i=1}^{1}(4^{i}\times(m_{1-i}^{c})^{2})$ $\displaystyle
T_{2}={}$ $\displaystyle 4(T_{1})+4^{8}+4\times m_{1}^{c}\times
4^{4}+4\times(m_{1}^{c})^{2}$ $\displaystyle={}$ $\displaystyle
4\left(4^{1}\times\sum_{i=0}^{1}4^{3i}+{\sum}_{i=1}^{1}(4^{1+i+1}\times
m_{i-1}^{c})+{\sum}_{i=1}^{1}(4^{i}\times(m_{1-i}^{c})^{2})\right)+4^{8}+4\times
m_{1}^{c}\times 4^{4}+4\times(m_{1}^{c})^{2}$ $\displaystyle={}$
$\displaystyle\left(4^{2}\times\sum_{i=0}^{1}4^{3i}+4^{8}\right)+\left(4{\sum}_{i=1}^{1}(4^{1+i+1}\times
m_{i-1}^{c})+4\times m_{1}^{c}\times
4^{4}\right)+\left(4{\sum}_{i=1}^{1}(4^{i}\times(m_{1-i}^{c})^{2})+4\times(m_{1}^{c})^{2}\right)$
$\displaystyle={}$ $\displaystyle
4^{2}\times\sum_{i=0}^{2}4^{3i}+{\sum}_{i=1}^{2}(4^{2+i+1}\times
m_{i-1}^{c})+{\sum}_{i=1}^{2}(4^{i}\times(m_{2-i}^{c})^{2})$ $\displaystyle
T_{3}={}$ $\displaystyle 4(T_{2})+4^{12}+4\times m_{2}^{c}\times
4^{6}+4\times(m_{2}^{c})^{2}$ $\displaystyle={}$ $\displaystyle
4\left(4^{2}\times\sum_{i=0}^{2}4^{3i}+{\sum}_{i=1}^{2}(4^{2+i+1}\times
m_{i-1}^{c})+{\sum}_{i=1}^{2}(4^{i}\times(m_{2-i}^{c})^{2})\right)+4^{12}+4\times
m_{2}^{c}\times 4^{6}+4\times(m_{2}^{c})^{2}$ $\displaystyle={}$
$\displaystyle\left(4^{3}\times\sum_{i=0}^{2}4^{3i}+4^{12}\right)+\left(4{\sum}_{i=1}^{2}(4^{2+i+1}\times
m_{i-1}^{c})+4\times m_{2}^{c}\times
4^{6}\right)+\left(4{\sum}_{i=1}^{2}(4^{i}\times(m_{2-i}^{c})^{2})+4\times(m_{2}^{c})^{2}\right)$
$\displaystyle={}$ $\displaystyle
4^{3}\times\sum_{i=0}^{3}4^{3i}+{\sum}_{i=1}^{3}(4^{3+i+1}\times
m_{i-1}^{c})+{\sum}_{i=1}^{3}(4^{i}\times(m_{3-i}^{c})^{2})$
$\displaystyle\vdots$ $\displaystyle T_{N}={}$
$\displaystyle\underbrace{4^{N}\times\sum_{i=0}^{N}4^{3i}}_{Q_{N}}+\underbrace{{\sum}_{i=1}^{N}(4^{N+i+1}\times
m_{i-1}^{c})}_{R_{N}}+\underbrace{{\sum}_{i=1}^{N}(4^{i}\times(m_{N-i}^{c})^{2})}_{S_{N}}$

We simplify for each $Q_{N},R_{N}$ and $S_{N}$.

Simplifying using geometric sum,

$\displaystyle Q_{N}$
$\displaystyle=4^{N}\times{\sum}_{i=0}^{N}4^{3i}=4^{N}\times\dfrac{(4^{3(N+1)}-1)}{4^{3}-1}=\dfrac{4^{N}}{63}(64\times
4^{3N}-1).$ (1) Using Lemma 1, $\displaystyle R_{N}$
$\displaystyle={\sum}_{i=1}^{N}(4^{N+i+1}\times m_{i-1}^{c})$
$\displaystyle={\sum}_{i=1}^{N}\left(\frac{4^{N+i+1}}{6}\times\dfrac{4^{i}(4^{i}-1)}{6}\right)=4^{N+1}\times{\sum}_{i=1}^{N}\dfrac{4^{3i}-4^{2i}}{6}=\dfrac{4^{N+1}}{6}\times\left({\sum}_{i=1}^{N}4^{3i}-{\sum}_{i=1}^{N}4^{2i}\right).$
Again, simplify using geometric sum, $\displaystyle R_{N}$
$\displaystyle=\dfrac{4^{N+1}}{6}\left(\dfrac{4^{3}(4^{3N}-1)}{4^{3}-1}-\dfrac{4^{2}(4^{2N}-1)}{4^{2}-1}\right)$
$\displaystyle=\dfrac{4^{N+1}}{1890}(320\times(4^{3N}-1)-336\times(4^{2N}-1))$
$\displaystyle=\dfrac{4^{N+1}}{1890}(320\times 4^{3N}-336\times 4^{2N}+16).$
(2) Lastly, $\displaystyle S_{N}$
$\displaystyle={\sum}_{i=1}^{N}(4^{i}\times(m_{N-i}^{c})^{2})$ By Lemma 1,
$\displaystyle S_{N}$
$\displaystyle={\sum}_{i=1}^{N}\left(4^{i}\times\left(\frac{4^{N-i+1}(4^{N-i+1}-1)}{6}\right)^{2}\right)$
$\displaystyle={\sum}_{i=1}^{N}\left(4^{i}\times\frac{4^{2N-2i+2}\left(4^{N-i+1}-1\right)^{2}}{36}\right)$
$\displaystyle=\dfrac{4^{N+1}}{9}{\sum}_{i=1}^{N}4^{N-i}\left(4^{2N-2i+2}-2\times
4^{N-i+1}+1\right)$
$\displaystyle=\dfrac{4^{N+1}}{9}\left({\sum}_{i=1}^{N}4^{3N-3i+2}-2\times{\sum}_{i=1}^{N}4^{2N-2i+1}+{\sum}_{i=1}^{N}4^{N-i}\right)$
$\displaystyle=\dfrac{4^{N+1}}{9}\left(16\times{\sum}_{i=1}^{N}4^{3(N-i)}-8\times{\sum}_{i=1}^{N}4^{2(N-i)}+{\sum}_{i=1}^{N}4^{N-i}\right).$
Simplify using geometric sum, $\displaystyle S_{N}$
$\displaystyle=\dfrac{4^{N+1}}{9}\left(16\times\frac{(4^{3N}-1)}{63}-8\times\frac{(4^{2N}-1)}{15}+\frac{(4^{N}-1)}{3}\right)$
$\displaystyle=\frac{4^{N+1}}{2835}\left(80\times(4^{3N}-1)-168\times(4^{2N}-1)+105\times(4^{N}-1)\right)$
$\displaystyle=\frac{4^{N+1}}{2835}\left(80\times 4^{3N}-168\times
4^{2N}+105\times 4^{N}-17\right).$ (3)

Thus,

$\displaystyle T_{N}={}$ $\displaystyle\dfrac{4^{N}}{63}(64\times
4^{3N}-1)+\dfrac{4^{N+1}}{1890}(320\times 4^{3N}-336\times 4^{2N}+16)$
$\displaystyle+\frac{4^{N+1}}{2835}\left(80\times 4^{3N}-168\times
4^{2N}+105\times 4^{N}-17\right)$ $\displaystyle={}$
$\displaystyle\dfrac{4^{N}}{5670}(10240\times 4^{3N}-5376\times
4^{2N}+840\times 4^{N}-34).$

∎

###  3.1 Counting $C_{4}$ in the theta graph $\Theta_{2,2,2}$

In this section, we construct the nested blow-up graph of the theta graph
$\Theta_{2,2,2}$ by replacing each vertex $v_{i}$ in $\Theta_{2,2,2}$ with a
$\Theta_{2,2,2}$.

Recall that the $N$ level nested blow-up of a graph $G$ is denoted $G_{N}$. We
compute the number of induced $C_{4}$ in $\Theta_{2,2,2}$. We denote $T_{N}$
as the number of induced $C_{4}$s and $n_{N}$ as the number of distinct
vertices in $G_{N}$ respectively. Figure 5 gives the theta graph
$\Theta_{2,2,2}$.

Figure 5: Illustration of theta graph $\Theta_{2,2,2}$.

######  Lemma 2.

Let $G=\Theta_{2,2,2}$. The number of non-edges $m^{c}_{N}$ in each level of
the blow-up graph $G_{N}$ is given by

$m_{N}^{c}=4\cdot 5^{N}\sum^{N}_{i=0}5^{i}=5^{N}(5^{N+1}-1).$

###### Proof.

We prove the equation by induction.
Base case: When $N=0$, there are $m^{c}_{0}=5^{0}(5^{1}-1)=4$ non-edges which
is precisely the number of non-edges in a $\Theta_{2,2,2}$.
Assume the induction hypothesis that for a particular $N$, the single case
$n=N$ holds, that is,

$\displaystyle m_{N}^{c}$ $\displaystyle=4\cdot
5^{N}\sum^{N}_{i=0}5^{i}=5^{N}(5^{N+1}-1).$ The number of non-edges in
$G_{N+1}$ is $\displaystyle m_{N+1}^{c}$
$\displaystyle=\binom{5^{N+2}}{2}-|E(G_{N+1})|$ The number of edges in $G_{N}$
is calculated as follows: There are 5 vertices and 6 edges in
$\Theta_{2,2,2}$. At the level $N$ blow-up, there are $6\cdot 5^{N}$ edges in
each blob, and also $6\cdot 5^{N+1}$ edges between blobs, thus giving the term
$|E(G_{N})|=6\cdot 5^{N}{\sum}_{i=0}^{N}5^{i}$. Thus, $\displaystyle
m_{N+1}^{c}$ $\displaystyle=\binom{5^{N+2}}{2}-6\cdot
5^{N+1}{\sum}_{i=0}^{N+1}5^{i}$
$\displaystyle=\dfrac{5^{N+2}(5^{N+2}-1)}{2}-\dfrac{(6\cdot
5^{N+1})(5^{N+2}-1)}{4}$ $\displaystyle=5^{N+1}(5^{N+2}-1).$

Since both the base case and inductive step has been proven as true, thus by
mathematical induction $m^{c}_{N}$ holds for all $N$. ∎

We state the following theorem.

######  Theorem 2.

The nested blow-up graph of a $\Theta_{2,2,2}$ has precisely
$T_{N}=\frac{5^{N}}{1240}(6300\times 5^{3N}-2945\times 5^{2N}+372\times
5^{N}-3877)$ induced subgraphs isomorphic to $C_{4}$.

###### Proof.

First, we will show that,

$T_{N}=\begin{cases}5\times(T_{N-1})+3\times(5^{N})^{4}+6\times(m^{c}_{N-1})^{2}+9\times
m^{c}_{N-1}\times(n_{N-1})^{2},&n>0\\\ 3&n=0.\end{cases}$

Since $G_{0}=\Theta_{2,2,2}$, we have $T_{0}=3$. We show that we can obtain
$T_{N}$ from $T_{N-1}$ and prove each term from $T_{N}$ respectively.

Note that at level $N$ blow-up, we replace each of the $5^{N+1}$ vertices in
$G_{N-1}$ with $G_{0}$, this contributes to $5\times T_{N-1}$ induced copies
of $C_{4}$ at $G_{N}$. Thus giving the first term $5\times T_{N-1}$.

$B_{1}$$B_{2}$$B_{3}$$B_{4}$$B_{5}$ Figure 6: The $\Theta_{2,2,2}$ graph
containing 5 blobs, with each blob labelled respectively.

We know that $G_{N}$ has five blobs isomorphic to $G_{N-1}$. For simplicity of
the rest of the proof, we will refer to the blobs as labelled
$B_{1},B_{2},B_{3},B_{4}$ and $B_{5}$, see Figure 6.

We select 4 of these 5 blobs and one vertex from each blob. There are five
ways to select 4 blobs, but only 3 of these induce copies of $C_{4}$. Each of
these three combinations of blobs contribute to 3 $C_{4}$ in $G_{N}$. There
are $5^{N}$ vertices in each $G_{N-1}$ which gives $3\times(5^{N})^{4}$
choices. This results in the second term $3\times(5^{N})^{4}$.

For the third term $6\times(m^{c}_{N-1})^{2}$, we choose 1 blob. We choose a
non-edge from this blob and another non-edge from any adjacent blob. There are
six pairs of blobs that are adjacent, thus resulting in the term
$6\times(m^{c}_{N-1})^{2}$.

Finally, we pick a non-edge in a blob $B$ and two vertices, one from each of
two different blobs each adjacent to $B$. There are nine ways that we can
select these (refer to Figure 6), namely, $\\{B_{1},B_{2},B_{5}\\}$,
$\\{B_{1},B_{3},B_{5}\\}$, $\\{B_{1},B_{4},B_{5}\\}$,
$\\{B_{1},B_{2},B_{3}\\}$, $\\{B_{1},B_{2},B_{4}\\}$,
$\\{B_{1},B_{3},B_{4}\\}$, $\\{B_{2},B_{3},B_{5}\\}$,
$\\{B_{2},B_{4},B_{5}\\}$, and $\\{B_{3},B_{4},B_{5}\\}$.

Each combination has $m^{c}_{N-1}$ choices of a non-edge from one blob and
then a choice of a vertex from the $n_{N-1}$ vertices in each of the other two
blobs. This results in the term $9\times m^{c}_{N-1}\times(n_{N-1})^{2}$. We
now expand and simplify $T_{N}$ to find the recurrence relation for $T_{N}$.
Expanding the first few terms, we obtain:

$\displaystyle T_{0}={}$ $\displaystyle 3$ $\displaystyle T_{1}={}$
$\displaystyle 5(T_{0})+3\times 5^{4}+6\times(m_{0}^{c})^{2}+9\times
m_{0}^{c}\times 5^{2}={}3\cdot(5+5^{4})+6\times(m_{0}^{c})^{2}+9\times
5^{2}\times m_{0}^{c}$ $\displaystyle={}$ $\displaystyle 3\cdot
5^{1}\times{\sum}_{i=0}^{1}5^{3i}+6\times{\sum}_{i=1}^{1}(5^{i-1}\times(m_{1-i}^{c})^{2})+9\times{\sum}_{i=1}^{1}(5^{1+i}\times
m_{i-1}^{c})$ $\displaystyle T_{2}={}$ $\displaystyle 5(T_{1})+3\times
5^{8}+6\times(m_{1}^{c})^{2}+9\times m_{1}^{c}\times 5^{4}$ $\displaystyle={}$
$\displaystyle 5\left(3\cdot
5^{1}\times{\sum}_{i=0}^{1}5^{3i}+6\times{\sum}_{i=1}^{1}(5^{i-1}\times(m_{1-i}^{c})^{2})+9\times{\sum}_{i=1}^{1}(5^{1+i}\times
m_{i-1}^{c})\right)+3\times 5^{8}+6\times(m_{1}^{c})^{2}+9\times
m_{1}^{c}\times 5^{4}$ $\displaystyle={}$ $\displaystyle\left(3\cdot
5^{2}{\sum}_{i=0}^{1}5^{3i}+3\cdot 5^{8}\right)+\left(6\cdot
5{\sum}_{i=1}^{1}(5^{i-1}\times(m_{1-i}^{c})^{2})+6\cdot(m_{1}^{c})^{2}\right)$
$\displaystyle+\left(9\cdot 5{\sum}_{i=1}^{1}(5^{1+i}\times
m_{i-1}^{c})+9\cdot m_{1}^{c}\cdot 5^{4})\right)$ $\displaystyle={}$
$\displaystyle 3\cdot
5^{2}\times{\sum}_{i=0}^{2}5^{3i}+6\times{\sum}_{i=1}^{2}(5^{i-1}\times(m_{2-i}^{c})^{2})+9\times{\sum}_{i=1}^{2}(5^{2+i}\times
m_{i-1}^{c})$ $\displaystyle T_{3}={}$ $\displaystyle 5(T_{2})+3\times
5^{12}+6\times(m_{2}^{c})^{2}+9\times m_{2}^{c}\times 5^{6}$
$\displaystyle={}$ $\displaystyle 5\left(3\cdot
5^{2}\times{\sum}_{i=0}^{2}5^{3i}+6\times{\sum}_{i=1}^{2}(5^{i-1}\times(m_{2-i}^{c})^{2})+9\times{\sum}_{i=1}^{2}(5^{2+i}\times
m_{i-1}^{c})\right)+3\times 5^{12}+6\times(m_{2}^{c})^{2}+9\times
m_{2}^{c}\times 5^{6}$ $\displaystyle={}$ $\displaystyle\left(3\cdot
5^{3}{\sum}_{i=0}^{2}5^{3i}+3\cdot 5^{12}\right)+\left(6\cdot
5{\sum}_{i=1}^{2}(5^{i-1}\times(m_{2-i}^{c})^{2})+6\cdot(m_{2}^{c})^{2}\right)$
$\displaystyle+\left(9\cdot 5{\sum}_{i=1}^{2}(5^{2+i}\times
m_{i-1}^{c})+9\cdot m_{2}^{c}\cdot 5^{6}\right)$ $\displaystyle={}$
$\displaystyle 3\cdot
5^{3}\times{\sum}_{i=0}^{3}5^{3i}+6\times{\sum}_{i=1}^{3}(5^{i-1}\times(m_{3-i}^{c})^{2})+9\times{\sum}_{i=1}^{3}(5^{3+i}\times
m_{i-1}^{c})$ $\displaystyle\vdots$ $\displaystyle T_{N}={}$
$\displaystyle\underbrace{3\cdot
5^{N}\times{\sum}_{i=0}^{N}5^{3i}}_{Q_{N}}+\underbrace{6\times{\sum}_{i=1}^{N}(5^{i-1}\times(m_{N-i}^{c})^{2})}_{R_{N}}+\underbrace{9\times{\sum}_{i=1}^{N}(5^{N+i}\times
m_{i-1}^{c})}_{S_{N}}.$

We simplify for each $Q_{N},R_{N}$ and $S_{N}$. Simplifying using geometric
sum,

$\displaystyle Q_{N}$ $\displaystyle=3\cdot
5^{N}\times{\sum}_{i=0}^{N}5^{3i}=3\cdot
5^{N}\times\dfrac{5^{3}(5^{3N}-1)}{5^{3}-1}=\dfrac{3\cdot
5^{N}}{124}(125\times 5^{3N}-1).$ (4) Using Lemma 2, $\displaystyle R_{N}$
$\displaystyle=6\times{\sum}_{i=1}^{N}(5^{i-1}\times(m_{N-i}^{c})^{2})$
$\displaystyle=6\times{\sum}_{i=1}^{N}(5^{i-1}\times(5^{N-i}(5^{N-i+1}-1))^{2})$
$\displaystyle=6\cdot
5^{N}\left(5\cdot{\sum}_{i=1}^{N}5^{3(N-i)}-2\cdot{\sum}_{i=1}^{N}5^{2(N-i)}+\frac{1}{5}\cdot{\sum}_{i=1}^{N}5^{N-i}\right).$
Again, simplify using geometric sum, $\displaystyle R_{N}$
$\displaystyle=6\cdot
5^{N}\left(5\times\frac{5^{3N}-1}{5^{3}-1}-2\times\frac{5^{2N}-1}{5^{2}-1}+\frac{1}{5}\times\frac{5^{N}-1}{5-1}\right)$
$\displaystyle=\dfrac{5^{N}}{620}(150\times 5^{3N}-310\times 5^{2N}+186\times
5^{N}-26).$ (5) Lastly, $\displaystyle S_{N}$
$\displaystyle=9\times{\sum}_{i=1}^{N}(5^{N+i}\times m_{i-1}^{c}).$ By Lemma
2, $\displaystyle S_{N}$
$\displaystyle=9\times{\sum}_{i=1}^{N}(5^{N+i}\times(5^{i-1}(5^{i}-1)))=9\cdot
5^{N}\times{\sum}_{i=1}^{N}(5^{3i-1}-5^{2i-1})=\dfrac{9\cdot
5^{N}}{5}\times\left({\sum}_{i=1}^{N}5^{3i}-{\sum}_{i=1}^{N}5^{2i}\right).$
Simplify using geometric sum, $\displaystyle S_{N}$
$\displaystyle=\dfrac{9\cdot
5^{N}}{5}\times\left(\dfrac{5^{3}(5^{3N}-1)}{5^{3}-1}-\dfrac{5^{2}(5^{2N}-1)}{5^{2}-1}\right)=\dfrac{3\cdot
5^{N}}{1240}(750\times 5^{3N}-775\times 5^{2N}+25).$ (6)

Thus,

$\displaystyle T_{N}={}$ $\displaystyle\dfrac{3\cdot 5^{N}}{124}(125\times
5^{3N}-1)+\dfrac{5^{N}}{620}(150\times 5^{3N}-310\times 5^{2N}+186\times
5^{N}-26)$ $\displaystyle+\dfrac{3\cdot 5^{N}}{1240}(750\times
5^{3N}-775\times 5^{2N}+25)$ $\displaystyle={}$
$\displaystyle\frac{5^{N}}{1240}(6300\times 5^{3N}-2945\times 5^{2N}+372\times
5^{N}-7).$ (7)

∎

##  4 Conclusion

In this paper, we gave exact counts of $C_{4}$s in two different graph
structures: (i) the nested blow-up graphs of $C_{4}$s and (ii) the theta graph
$\theta_{2,2,2}$. Previously, only bounds were found for the blow-ups of
graphs [14]. We improved the bounds to give the exact counts of such graphs.
In a general case, we can adapt a similar formula to construct the equations
for blow-up graphs of higher order $k$s, to find the exact counts of cycles of
higher order $k$. Future direction of this work could include finding a
generalised formula for any types of blow-up graphs, with _some_ cyclic
property.

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# Open charm phenomenology with a multi-stage approach to relativistic heavy-
ion collisions

Mayank Singh<EMAIL_ADDRESS>School of Physics & Astronomy, University of
Minnesota, Minneapolis, MN 55455, USA  Manu Kurian<EMAIL_ADDRESS>Department
of Physics, McGill University, 3600 University Street, Montreal, QC, H3A 2T8,
Canada  RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New
York 11973, USA  Sangyong Jeon<EMAIL_ADDRESS>Department of Physics,
McGill University, 3600 University Street, Montreal, QC, H3A 2T8, Canada
Charles Gale<EMAIL_ADDRESS>Department of Physics, McGill
University, 3600 University Street, Montreal, QC, H3A 2T8, Canada

###### Abstract

We study open charm flavor observables in Pb+Pb collision at
$\sqrt{s_{NN}}=2.76$ TeV within the MARTINI framework. The space-time
expansion of the quark-gluon plasma is described using the hydrodynamical
approach-MUSIC with IP-Glasma initial conditions. The model parameters,
including the viscous coefficients, were obtained from a recent Bayesian
model-to-data comparison. We evolve heavy quarks in this background using
Langevin dynamics while incorporating their collisional and radiative
processes in the medium. The sensitivity of charm observables to the IP-Glasma
initial state, bulk evolution, and centrality of the collision is studied. We
find that the elliptic flow of open charm flavor has a strong dependence on
the fluctuating initial conditions in addition to the strength of the
interaction of heavy quarks with the medium constituents. Within this
framework, the nuclear suppression factor and elliptic flow of D-mesons act as
efficient probes to study the initial stages of heavy-ion collisions,
transport coefficients associated with QGP medium as well as heavy quark
interactions.

_Introduction_ \- Heavy-ion collision experiments at the Relativistic Heavy
Ion Collider (RHIC) and Large Hadron Collider (LHC) provide an access to
strongly interacting matter: the Quark-Gluon Plasma (QGP). Relativistic
viscous hydrodynamics serves as an important framework to study the dynamical
evolution of the QGPGale  _et al._ (2013). The transport properties of QGP are
extracted from model to light flavor hadron data comparisons, and significant
effort has been devoted in this directionRomatschke and Romatschke (2007);
Heinz and Snellings (2013); Ryu  _et al._ (2015); Jaiswal and Roy (2016).
Recently, Bayesian analysis has been used to do a systematic extraction of
shear and bulk viscosities of the medium and their uncertainty quantification
Bernhard  _et al._ (2019); Everett  _et al._ (2021); Nijs  _et al._ (2021);
Heffernan  _et al._ (2023).

Heavy flavor quarks provide additional probes to study the properties of QGP
van Hees  _et al._ (2006); Das  _et al._ (2009); He  _et al._ (2013a);
Andronic  _et al._ (2016); Aarts  _et al._ (2017); Song  _et al._ (2020a);
Mustafa (2005); Sun  _et al._ (2019). They are largely created in the initial
stages of the collision, and pass through the QGP while interacting with the
light flavor quarks and gluons. Thermal production of heavy quarks in the QGP
medium is expected to be negligible because of their larger mass compared to
the temperature scale of the medium Dong and Greco (2019). Generally, heavy
flavor particles are not treated as part of the medium as the thermalization
time is longer than the QGP lifetime. Their dynamics can be described as
Brownian motion in the medium and can be studied within the Langevin or the
Fokker-Planck frameworks Golam Mustafa  _et al._ (1998); Moore and Teaney
(2005); Uphoff  _et al._ (2011); Das  _et al._ (2014); Cao  _et al._ (2016);
Li  _et al._ (2019). Nuclear suppression factor $R_{AA}$ and elliptic flow
$v_{2}$ are the key observables associated with the heavy flavor particles at
the RHIC and LHC energies. Several studies have been done to estimate $R_{AA}$
and $v_{2}$ by modeling the Brownian motion of the heavy quarks in the medium
Akamatsu  _et al._ (2009); He  _et al._ (2012); Cao  _et al._ (2013); Alberico
_et al._ (2013); Song  _et al._ (2015); Li  _et al._ (2018); van Hees  _et
al._ (2008) and to extract the momentum and temperature behavior of heavy
quark transport coefficients from these observables in heavy-ion collision
experiments Xu  _et al._ (2018); Scardina  _et al._ (2017); Beraudo  _et al._
(2018); Cao  _et al._ (2019). Notably, the inclusion of inelastic interactions
of heavy quarks in the medium on top of the elastic collisions reduces the gap
between experimental and theoretical observations Wicks  _et al._ (2007).

The evolution history of the collision event is essential to model the heavy
quark dynamics in the expanding medium. Heavy quark production and transport
have been widely studied in the static limit and in expanding fireball models.
Some efforts have been made to study the heavy flavor dynamics in evolving
medium within 1+1 Bjorken hydrodynamics as well as higher dimension
hydrodynamics Young  _et al._ (2012); He  _et al._ (2013b); Sarkar  _et al._
(2018); Thakur  _et al._ (2020); Prakash  _et al._ (2021). Most analyses
utilized smooth initial conditions for the hydrodynamical evolution of the QGP
medium through which heavy quarks traverse. Advances in hydrodynamical models
to describe medium expansion seem to have a significant impact on the heavy
quark observables Gossiaux  _et al._ (2011); Song  _et al._ (2020b); Fan  _et
al._ (2023). It is also known that the initial state fluctuation has a
significant influence on the light flavor jet energy loss Rodriguez  _et al._
(2010); Zhang  _et al._ (2013).

In this work, we employ the state-of-the-art hydrodynamical model of QGP to
study the charm observables. We consider the evolution history of a Pb+Pb
collision event at 2.76 TeV energy with the IP-Glasma initial state Schenke
_et al._ (2012a, b); McDonald  _et al._ (2017). IP-Glasma is a very successful
dynamical model which describes a variety of observables in heavy-ion
collisions. The initial fluctuating color configuration in the heavy-ion
nuclei that are approaching at high velocity can be determined within IP-Sat
approach Bartels  _et al._ (2002); Kowalski and Teaney (2003) combined with
the Yang-Mills equations Krasnitz and Venugopalan (1999). Its evolution also
follows from solving the Classical Yang-Mills equations. The Glasma
distributions, which are obtained event-by-event, act as the input for the
hydrodynamical evolution. We used the recently updated shear and bulk viscous
coefficients obtained from the Bayesian model-to-data analysis Heffernan  _et
al._ (2023).

The charm dynamics are encoded in the drag and diffusion coefficients in the
Langevin equation. The Langevin equation is solved within MARTINI Schenke  _et
al._ (2009); Young  _et al._ (2012). We studied open charm observables within
three different setups of drag and diffusion coefficients. This analysis is an
up-to-date study of the heavy flavor nuclear suppression factor and elliptic
flow using the recent developments in the hydrodynamical description of QGP
and drag and diffusion coefficients of the charm quarks.

_A hybrid model for heavy quarks_ \- Modelling of heavy flavor evolution in
collisions can be divided into three distinct stages: heavy quark initial
production, its evolution in the hydrodynamized expanding medium, and the
hadronization process. In the current analysis, we employ PYTHIA8.2 Sjöstrand
_et al._ (2015) for perturbative production of charm quarks by sampling
6-dimensional momentum distributions of $Q\bar{Q}$ systems. We allow the
gluons to split into $c\bar{c}$ pairs, but the medium-induced modification of
the gluon splitting rate is not accounted for. As the nucleons are bound in a
heavy nucleus, the parton distribution functions (nPDFs) will be modified. To
take into account the nuclear shadowing effect, EPS09 nuclear parton
distribution functions are used Eskola  _et al._ (2009). Isospin effects are
accounted for by sampling $p+p$, $p+n$ and $n+n$ collisions. A finite
thermalization time $\tau_{0}$ in the heavy-ion collision is considered, and
the evolution of charm quarks during this time is governed by the equation of
motion with the zero-temperature Cornell potential Young  _et al._ (2012).

The QGP medium is initialized using the IP-Glasma model Schenke  _et al._
(2012a, b); McDonald  _et al._ (2017) and evolved using the viscous
hydrodynamical approach MUSIC Schenke  _et al._ (2010, 2011). The lattice QCD
equation of state (EoS) from the hotQCD collaboration Bazavov  _et al._ (2014)
at high temperature is smoothly matched to the hadron resonance gas EoS at low
temperatures and is incorporated in the framework. The first thorough Bayesian
model-to-data comparison of relativistic heavy-ion collision measurements
using a hybrid model combining viscous expansion (MUSIC), evolution to
particlization (iS3D) McNelis  _et al._ (2021), and particle transport (SMASH)
Weil  _et al._ (2016) with IP-Glasma initial conditions has been recently
presented for four different model choices in Heffernan  _et al._ (2023). We
chose the model with Grad’s 14-moment viscous correction with constant shear
viscosity to entropy density ratio and a temperature-dependent bulk viscosity
profile. In the present analysis, we have used the maximum a posteriori
estimates of all model parameters, including the viscous coefficients, from
this study.

The dynamics of heavy quarks in the QGP medium depend upon its radiative and
elastic interactions with the constituents in the medium. The strength of
interactions of heavy quarks in the medium can be quantified in terms of drag
and diffusion coefficients. In the local rest frame of the medium, the heavy
quark motion can be studied numerically using the discrete version of the
Langevin equations Moore and Teaney (2005); Das  _et al._ (2014),

$\displaystyle dp_{i}=-A_{i}\,dt+C_{ij}\rho_{j}\sqrt{dt},$ (1)

where $dp_{i}$ denotes the change in momentum in time interval $dt$. The drag
force $A_{i}$ and covariance matrix $C_{ij}$ are defined as follows,

$\displaystyle A_{i}=p_{i}A(|{\bf p}|^{2},T),$ (2) $\displaystyle
C_{ij}=\sqrt{2B_{0}}\left(\delta_{ij}-\frac{p_{i}p_{j}}{|{\bf
p}|^{2}}\right)+\sqrt{2B_{1}}\frac{p_{i}p_{j}}{|{\bf p}|^{2}},$ (3)

where $A$ denotes the drag coefficient and the quantities $B_{0}$ and $B_{1}$
represent the transverse and longitudinal momentum diffusion coefficients,
respectively. The drag force describes the heavy quark average momentum
transfer due to the interactions, whereas the matrix $C_{ij}$ quantifies the
stochastic force by using Gaussian-normal distributed random variable
$\rho_{j}$Das  _et al._ (2014). The dependence of drag and diffusion on heavy
quark momentum and temperature of the medium can be studied within
relativistic transport theory. The Langevin dynamics of the heavy quarks is
coupled with the expanding QGP medium and solved within MARTINI as follows:

  * $-$

Find the fluid four-velocity and temperature at the space-time location of the
charm quark from MUSIC

  * $-$

Boost the charm momentum to the fluid local rest frame and evolve the charm
three momenta to the next time step using Langevin equations

  * $-$

Boost the charm momentum back to the lab frame and update the charm position
after the time step

The heavy quark transport coefficients are the key input parameters that
quantify the interaction strength of heavy quarks with the medium. The
transport coefficients can be derived in perturbative QCD by including
scattering and radiative processes Svetitsky (1988); Mustafa (2005). The
specific interactions are encoded in the matrix elements. It has been seen
that using Debye mass as infrared (IR) regulator ($\mu_{IR}$) in the gluon
propagator for t-channel interaction and a fixed coupling constant, pQCD
matrix elements are not able to describe the experimental data associated with
the heavy flavor particles Rapp and van Hees (2010); Beraudo  _et al._ (2018).
As these parameters are the sources of uncertainty in the estimation of heavy
quark transport coefficients, the IR regulator and coupling constant are
determined in the analysis by physical arguments as described in Gossiaux and
Aichelin (2008). In the present analysis, the conventional choice of IR
regulator, the Debye mass, is replaced with a realistic hard thermal loop
(HTL) parameterization of the IR regulator. Further, an effective coupling
constant ($\alpha_{\text{eff}}$) that embeds non-perturbative dynamics is
employed in the study. The behaviour of $\alpha_{\text{eff}}$ is obtained from
the analysis of $e^{+}e^{-}$ annihilation Mattingly and Stevenson (1994) and
decay of $\tau$ leptons Brodsky  _et al._ (2003), and has the following form
Gossiaux and Aichelin (2008),

$\alpha_{\text{eff}}(R^{2})=\dfrac{4\pi}{\beta_{0}}\Bigg{\\{}\begin{array}[]{lr}L_{-}^{-1},&R^{2}<0\\\
\frac{1}{2}-\pi^{-1}\text{arctan}(L_{+}/\pi),&R^{2}>0\end{array}$ (4)

where $R$ is the relevant energy scale, $\beta_{0}=11-\frac{2}{3}N_{f}$ with
$N_{f}$ as the number of favors and $L_{\pm}=\ln{(\pm R^{2}/\Lambda^{2})}$
with QCD parameter chosen as $\Lambda=0.2$ GeV. With these parameterizations,
the propagator with bare coupling $\alpha_{s}$ is modified for $t-$channel
gluon exchange processes (heavy quark-thermal medium interaction) as,

$\displaystyle\frac{\alpha}{t}\rightarrow\frac{\alpha_{\text{eff}}\,(t)}{t-\mu^{2}_{IR}(t,T)},$
(5)

where $t$ is the Mandelstam variable and
$\mu^{2}_{IR}(t,T)=\kappa\,4\pi\Big{(}1+\frac{N_{f}}{6}\Big{)}\alpha_{\text{eff}}\,(t)T^{2}$
with $\kappa=0.2$. IR regulator is not required for other channels and
coupling constant is fixed as
$\alpha\rightarrow\alpha_{\text{eff}}\,(s-m^{2}_{HQ})$ and
$\alpha\rightarrow\alpha_{\text{eff}}\,(u-m^{2}_{HQ})$ for $s$ and $u-$
channels, respectively such that $s=m^{2}_{HQ}$ and $u=m^{2}_{HQ}$ denote the
maximal softness in the channels Gossiaux and Aichelin (2008). Here, $m_{HQ}$
is the mass of the heavy quark. For the charm quark, we took $m_{HQ}=1.25$
GeV.

Figure 1: Temperature dependence of spatial diffusion coefficient of charm
quark and comparison of the results with other approaches.

In this study, we consider three different setups to characterize the coupling
strength of heavy quarks in the QGP medium. (i) Setup I-Constant value of
$2\pi D_{s}T$: In the static limit of heavy quarks, the coupling of heavy
quarks in the QGP can be described with one physical parameter-the spatial
diffusion coefficient $D_{s}$, which is often treated as a phenomenological
parameter. For $\sqrt{s_{NN}}=2.76$ TeV energy collisions, we take $2\pi
D_{s}T=3.0$. (ii) Setup II-Temperature dependent heavy quark transport
coefficients: Employing the fluctuation-dissipation theorem, $D_{s}$ can be
expressed in terms of a drag coefficient in the limit $p\rightarrow 0$ as,
$D_{s}=\frac{T}{m_{HQ}A(p\rightarrow 0,T)}$. The momentum dependence of
$D_{s}$ is neglected in this scenario. The temperature dependence of $D_{s}$
for the collisional and radiative processes with the effective coupling and
modified IR regulator is depicted in fig. 1. For the pQCD elastic collisional
process, the value of $(2\pi T)D_{s}$ lies in the range of $30-40$ van Hees
and Rapp (2005); Kurian  _et al._ (2020), which is an order of magnitude
larger than that obtained from $N_{f}=0$ lattice result Banerjee  _et al._
(2012), $N_{f}=2+1$ lattice data Altenkort  _et al._ (2023) and quasiparticle
model (QPM) Scardina  _et al._ (2017) estimation. It is seen that the gluon
radiation of heavy quarks in the medium suppresses the $D_{s}$. This can be
understood from the fact that the heavy quark is experiencing more drag as
they lose energy through collisional and radiative processes while traveling
through the QGP medium. The effective coupling that incorporates the non-
perturbative effects and the HTL IR regulator seems to have significant
impacts on the temperature behavior of $D_{s}$. With the above choice of
parameters and the inclusion of the radiative process in the analysis, it is
observed that $(2\pi T)D_{s}\approx 2-7$. (iii) Setup III-Temperature and
momentum dependent heavy quark transport coefficients: For a heavy quark with
a finite momentum, its dynamics in the QGP medium are described with
parameters, $A(p,T)$, $B_{0}(p,T)$, $B_{1}(p,T)$ where, in general, $B_{0}\neq
B_{1}$ (see eqs. 2 and 3). The momentum and temperature dependence of the
heavy quark drag and diffusion coefficients is described in eqs. 6-8. It is
seen that the drag coefficient decreases with an increase in heavy quark
momentum, whereas the trend is quite the opposite for momentum diffusion
coefficients. The transverse diffusion coefficient $B_{0}$ increases with
momentum and saturates at higher momentum. However, the coefficient $B_{1}$
has a sharp rise with an increase in heavy quark momentum, which indicates
large random kicks to the heavy quark. The fluctuation-dissipation relation is
enforced for the longitudinal diffusion coefficient to ensure the
equilibration of heavy quarks in the medium. The details of the derivation of
drag and diffusion coefficient are given in appendix A. We have also analyzed
the viscous effects to the heavy quark transport coefficients. It is seen that
viscous effects have no visible impact on the temperature behavior of $D_{s}$,
especially in the high-temperature regime Kurian  _et al._ (2020).

In MARTINI, Peterson fragmentation is employed to describe the heavy quark
hadronization process. For the open heavy flavor mesons, heavy quark fragments
into a meson, and the fragmentation function estimates the fractional momentum
of the resulting hadron. Quarkonium is another possible final state for the
heavy quarks, and MARTINI separates out this final state from open heavy
flavor mesons using a three-step algorithm as described in detail in Young
_et al._ (2012).

_Results and discussions_ -

Figure 2: Nuclear suppression factor and elliptic flow of D mesons in three
different setups. Experimental data of D meson $R_{AA}$ and $v_{2}$ are from
the ALICE collaboration, Refs. Abelev  _et al._ (2013) and Adam  _et al._
(2016), respectively. The nuclear suppression factor for setups I and II are
almost identical.

We evaluate the $R_{AA}$ and $v_{2}$ of D mesons. The elliptic flow harmonic
$v_{2}$ was evaluated using the event plane method where the azimuthal angle
of D meson $\phi_{D}$ is correlated with the second order event-plane angle
$\Psi_{2}$. In experiments, the event plane angle is determined by the
azimuthal distribution of all charged hadrons. We used the initial state
spatial anisotropy to determine the event-plane angle. In the studied
centrality class, the spatial anisotropy angle and the event plane angle are
strongly correlated.

We compared the results with the available experimental data at $30-50\%$. The
nuclear suppression factor $R_{AA}$ and elliptic flow $v_{2}$ of the D-mesons
are shown in fig. 2. We observe that the estimation with momentum and
temperature-dependent charm quark transport coefficients have a better
agreement with the available data for $R_{AA}$ in comparison with the other
two setups. In contrast, the setup underestimates the D meson $v_{2}$. A
recent study Das  _et al._ (2015) has predicted that the temperature
dependence of heavy quark interaction strength plays a vital role in the
simultaneous description of both $R_{AA}$ and $v_{2}$ of D meson at the RHIC
energy. However, there are still mismatches between the calculations and
measurements, especially at the LHC energies. Notably, none of the models
could explain the enhancement in $v_{2}$ around $p_{T}=10$ GeV. This could be
due to the uncertainties of heavy quark interaction in the medium and heavy
flavor hadronization process, especially in the low $p_{T}$ regime where the
coalescence mechanism may have an impact on the observables Fries  _et al._
(2008).

Figure 3: Nuclear suppression factor and elliptic flow of D mesons in setup-II
in two different centralities with event-by-event initialized and smooth hydro
backgrounds.

To quantify the impact of fluctuating IP-Glasma initial state on heavy quark
observables, we have compared the results from event-by-event initialized
calculations with those from smooth initial conditions. The smooth initial
profiles are obtained from the optical Glauber model for impact parameters 3.5
fm and 10 fm, which roughly correspond to the 0-10% and 30-50% centrality
bins. All other parameters were held fixed.

The impact of fluctuating initial conditions on D meson $v_{2}$ is illustrated
in fig. 3. At low $p_{T}$, the fluctuating initial condition seems to have a
significant influence on the D-meson $v_{2}$ for $30-50\%$ centrality. The
effect of fluctuating initial conditions can be understood as a convolution of
two distinct effects. Fluctuations increase local pressure gradients and
enhance flow. That will lead to an increase in $v_{2}$. However, fluctuations
also increase decorrelation between the event planes of light flavor and heavy
flavor mesons. As the two are produced by different mechanisms in different
stages of evolution, their event plane angles are generally not identical.
Heavy flavor meson $v_{2}$ is measured by taking its projection on the event
plane determined by charged hadrons, which is dominated by the light flavor
mesons. This increased decorrelation suppresses $v_{2}$. The net effect is a
combination of two factors. As the charm quarks are much heavier than the
background medium, the enhancement in $v_{2}$ from the increased flow is more
than compensated by the decorrelation. This effect is opposite to that
observed in jets Noronha-Hostler  _et al._ (2016).

_Summary_ \- In this paper, a hybrid framework is developed to study the
evolution of heavy flavor by incorporating the recent developments in initial
state dynamics and viscous QGP evolution in the relativistic heavy-ion
collisions in Pb+Pb collision at $\sqrt{s_{NN}}=2.76$ TeV. We introduce
fluctuating IP-Glasma initial states and viscous hydrodynamics tuned to a
global Bayesian analysis for the first time in a phenomenological study of the
charm quark. The heavy quark dynamics is described within the Langevin
approach in the expanding medium in which heavy quark coupling strength with
the medium is quantified in terms of its transport coefficients. We explored
the momentum and temperature dependence of the charm quark transport
coefficients due to the collisional and radiative energy loss of the heavy
quarks in the QGP medium, as well as its impact on the nuclear modification
factor $R_{AA}$ and the elliptic flow $v_{2}$ of D-mesons. Our results with
improved heavy quark dynamics with the latest developments in multistage
hybrid frameworks for the dynamical evolution of collisions demonstrate that
heavy flavor observables are influenced by the IP-Glasma initial state and
bulk evolution of the medium. We see that fluctuating initial conditions have
a significant effect on charm elliptic flow at low $p_{T}$. These effects are
the result of an increase in both flow and decorrelations. While enhanced flow
is the dominant effect for light jets, the event-plane decorrelation is more
important for charm quarks. This indicates that the heavier charm quark is
less susceptible to becoming part of the background flow than light quarks.
Further, we observe that the energy loss profiles of a charm quark and non-
perturbative effects in the QGP medium have a significant role in both
$R_{AA}$ and $v_{2}$ of D-mesons.

Looking into the future, it will be interesting to explore the influence of
pre-equilibrium interactions on heavy quark energy loss. These effects are
essential to maintain coherence in the theoretical description of heavy flavor
dynamics in heavy-ion collisions. Additionally, it is an important aspect to
take into account the uncertainties associated with the momentum and
temperature dependence of heavy quark transport coefficients to simultaneously
describe $R_{AA}$ and $v_{2}$ in Pb+Pb collisions at $2.76$ TeV. This tuning
can be achieved by utilizing a model-to-data comparison. We leave these
interesting aspects to the near future.

_Acknowledgements_ \- We thank Björn Schenke and Gojko Vujanovic for helpful
discussions and feedback. We acknowledge Matthew Heffernan and Nicolas Fortier
for their help with the IP-Glasma initial state files and with MUSIC
parameters. Numerical computations were done on the resources provided by the
Minnesota Supercomputing Institute (MSI) at the University of Minnesota and on
Beluga supercomputer at McGill University managed by Calcul Québec and Compute
Canada. M.S. is supported by the U.S. DOE Grant No. DE-FG02-87ER40328. M.K.
acknowledges a fellowship from the Fonds de recherche du Québec - Nature et
technologies (FRQNT), support from the Natural Sciences and Engineering
Research Council of Canada, and the Special Postdoctoral Researchers Program
of RIKEN. S.J. and C.G. are supported by the Natural Sciences and Engineering
Research Council of Canada.

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##  Appendix A Appendix A: temperature and momentum dependence of charm quark
transport coefficients

Figure 4: Drag coefficient A (top) and diffusion coefficient $B_{0}$ (bottom)
as a function of charm quark momentum in the local rest frame.

By employing the Landau approximation Landau , the collision integral in the
Boltzmann equation can be simplified and the transport coefficients can be
defined as,

$\displaystyle A=~{}\langle\langle 1\rangle\rangle-\dfrac{\langle\langle{\bf
p}\cdot{\bf p}^{\prime}\rangle\rangle}{|{\bf p}|^{2}},$ (6) $\displaystyle
B_{0}=\dfrac{1}{4}\bigg{[}\langle\langle{|{\bf
p}^{\prime}|}^{2}\rangle\rangle-\dfrac{\langle\langle({\bf p}\cdot{\bf
p}^{\prime})^{2}\rangle\rangle}{|{\bf p}|^{2}}\bigg{]},$ (7) $\displaystyle
B_{1}=\dfrac{1}{2}\bigg{[}\dfrac{\langle\langle({\bf p}\cdot{\bf
p}^{\prime})^{2}\rangle\rangle}{|{\bf p}|^{2}}-2\langle\langle{\bf p}\cdot{\bf
p}^{\prime}\rangle\rangle+|{\bf p}|^{2}\langle\langle
1\rangle\rangle\bigg{]},$ (8)

where $\langle\langle F(|{\bf p}^{\prime}|)\rangle\rangle$ denote the thermal
average of a function $F(|{\bf p}^{\prime}|)$ and depends upon the heavy quark
interaction process in medium.

For the elastic collisional process, $HQ(p)+l(q)\rightarrow
HQ(p^{\prime})+l(q^{\prime})$, where $l$ denotes light quarks or gluons,
$\langle\langle F(|{\bf p}^{\prime}|)\rangle\rangle$ is defined as,

$\displaystyle\langle\langle F(|{\bf p}^{\prime}|)$
$\displaystyle\rangle\rangle=\dfrac{1}{\gamma_{HQ}}\dfrac{1}{2E_{p}}\int{\dfrac{d^{3}{\bf
q}}{(2\pi)^{3}2E_{q}}}\int\frac{d^{3}{\bf
p^{\prime}}}{(2\pi)^{3}2E_{p^{\prime}}}$
$\displaystyle\times\int\frac{d^{3}{\bf
q^{\prime}}}{(2\pi)^{3}2E_{q^{\prime}}}(2\pi)^{4}\delta^{4}(p+q-p^{\prime}-q^{\prime})$
$\displaystyle\times\sum|{\mathcal{M}}_{2\rightarrow
2}|^{2}f_{g/q}({E_{q}})\Big{(}1\pm f_{g/q}(E_{q^{\prime}})\Big{)}F(|{\bf
p}^{\prime}|),$ (9)

where $\gamma_{HQ}$ as the statistical degeneracy factor of heavy quark and
$|{\mathcal{M}}_{2\rightarrow 2}|$ describes interaction amplitude of the
heavy quark-thermal particles elastic scattering process. Here, $f_{g/q}$ is
the distribution function of thermal particles in the evolving medium.

For the inelastic ($2\rightarrow 3$) process, $HQ(p)+l(q)\rightarrow
HQ(p^{\prime})+l(q^{\prime})+g(k^{\prime})$, with
$k^{\prime}\equiv(E_{k^{\prime}},{\bf k^{\prime}_{\perp}},k^{\prime}_{z})$ as
the four-momentum of the emitted soft gluon by the heavy quark in the final
state, the thermal averaged $F(|{\bf p}^{\prime}|)$ takes the form as Mazumder
_et al._ (2014),

$\displaystyle\langle\langle F(|{\bf p}^{\prime}|)$
$\displaystyle\rangle\rangle=\dfrac{1}{\gamma_{HQ}}\frac{1}{2E_{p}}\int\frac{d^{3}{\bf
q}}{(2\pi)^{3}2E_{q}}\int\frac{d^{3}{\bf
p^{\prime}}}{(2\pi)^{3}2E_{p^{\prime}}}$
$\displaystyle\times\int\frac{d^{3}{\bf
q^{\prime}}}{(2\pi)^{3}2E_{q^{\prime}}}\int\frac{d^{3}{\bf
k^{\prime}}}{(2\pi)^{3}2E_{k^{\prime}}}\ (2\pi)^{4}$
$\displaystyle\times\delta^{(4)}(p+q-p^{\prime}-q^{\prime}-k^{\prime})\sum{|{\mathcal{M}}_{2\rightarrow
3}|^{2}}$ $\displaystyle\times f_{g/q}(E_{q})(1\pm f_{g/q}(E_{q^{\prime}}))\
(1+f_{g}(E_{k^{\prime}}))\ $
$\displaystyle\times\theta_{1}(E_{p}-E_{k^{\prime}})\
\theta_{2}(\tau-\tau_{F})\ F(|{\bf p}^{\prime}|),$ (10)

where $|{\mathcal{M}}_{2\rightarrow 3}|^{2}$ describes the matrix element
squared the radiative process Abir  _et al._ (2012). The theta function
$\theta_{1}(E_{p}-E_{k^{\prime}})$ imposes constraints on the heavy quark
initial energy and $\theta_{2}(\tau-\tau_{F})$ indicates that scattering time
$\tau$ is larger than the gluon formation time $\tau_{F}$ (Landau-Pomeranchuk-
Migdal Effect) Gyulassy and Wang (1994); Klein (1999).

Fig. 4 shows the drag and diffusion coefficients of charm quark as a function
of momentum while including the collisional and radiative processes with the
modified IR regulator and effective coupling.