# The fullerenes with a perfect star packing*** This work was supported in
part by the National Natural Science Foundation of China (grant no. 11901458
and 11871256) and by the Fundamental Research Funds for the Central
Universities (grant no. D5000200199).

Ling-Juan Shi Corresponding author.

(School of Software, Northwestern Polytechnical University,
Xi’an, Shaanxi 710072, P. R. China
E-mails<EMAIL_ADDRESS>
###### Abstract

A spanning subgraph of a graph $G$ is called a perfect star packing in $G$ if
every component of the spanning subgraph is isomorphic to the star graph
$K_{1,3}$. An efficient dominating set of graph $G$ is a vertex subset $D$ of
$G$ such that each vertex of $G$ not in $D$ is adjacent to exactly one vertex
from $D$ and any two vertices of $D$ are not adjacent in $G$. Fullerene graph
is a connected plane cubic graph with only pentagonal and hexagonal faces,
which is the molecular graph of carbon fullerene. Clearly, a perfect star
packing in a fullerene graph $G$ on $n$ vertices will exist if and only if $G$
has an efficient dominating set of cardinality $\frac{n}{4}$. The problem of
finding an efficient dominating set is algorithmically hard [2]. In this
paper, we give a characterization for a fullerene graph to own a perfect star
packing. And mainly show that it is necessary for a fullerene $G$ owning a
perfect star packing to have order being divisible by $8$. This answers an
open problem asked by Doslić et. al. and also shows that a fullerene graph
with an efficient dominating set has $8n$ vertices. By the way, we find some
counterexamples for the necessity of Theorem $14$ in [13] and list some
forbidden configurations to preclude the existence of a perfect star packing
of type $P0$.

Keywords: Fullerene graph; Perfect star packing; Efficient dominating set

##  1 Introduction

A chemical graph is a simple finite graph in which vertices denote the atoms
and edges denote the chemical bonds in underlying chemical structure. Perfect
matchings of a chemical graph correspond to kekulé structures of the molecule,
which feature in the calculation of molecular energies associated with
benzenoid hydrocarbon molecules [19]. Alternating sextet faces (sextet
patterns) also play a meaningful role in the prediction of molecular
stability, in particular, but not only, in benzenoid compounds. Although for
fullerenes, the two structures do not play the same role as in benzenoid
compounds, they received numerous attention in recent years, see [1, 3, 7, 12,
16, 20, 32, 33] etc.

A perfect matching in a graph $G$ may be viewed as a collection of subgraphs
of $G$, each of which is isomorphic to $K_{2}$, whose vertex sets partition
the vertex set of $G$. This is naturally generalized by replacing $K_{2}$ by
an arbitrary graph $H$. For a given graph $H$, an _$H$ -packing_ of $G$ is the
set of some vertex disjoint subgraphs, each of which is isomorphic to $H$.
From the optimization point of view, the maximum $H$-packing problem is to
find the maximum number of vertex disjoint copies of $H$ in $G$ called the
_packing number_. An $H$-packing in $G$ is called _perfect_ if it covers all
the vertices of $G$. If $H$ is isomorphic to $K_{2}$, the maximum (perfect)
$H$-packing problem becomes the familiar maximum (perfect) matching problem.
If $H$ is the cycle $C_{6}$ of length $6$, for a fullerene or a hexagonal
system $G$, the packing number is related to the Clar number (the maximum
number of mutually disjoint sextet patterns) of $G$. If $H$ is the star graph
$K_{1,3}$, it is the maximum star packing problem. If a $K_{1,3}$-packing
covers all the vertices of $G$, we call it being a _perfect star packing_. For
a given family $\mathcal{F}$ of graphs, an $H$-packing concept can also be
generalized to an $\mathcal{F}$-packing (we refer the reader to [29] for the
definition).

Packing in graphs is an effective tool as it has lots of applications in
applied sciences. $H$-Packing, is of practical interest in the areas of
scheduling [5], wireless sensor tracking [6], wiring-board design, code
optimization [21] and many others. Packing problems were already studied for
Carbon Nanotubes [27]. Packing lines in a hypercube have been studied in [14].
$H$-packing was determined for honeycomb [29] and hexagonal network [28]. For
representing chemical compounds or to problems of pattern recognition and
image processing, $P_{3}$-packing has some applications in chemistry [30].
Doslić et. al [13] have investigated which fullerene graphs allow perfect star
packings, and have considered generalized fullerene graphs and packings of
other graphs into classical and generalized fullerenes. They also listed
several open problems.

In the following section we introduce necessary preliminaries and characterize
the classical fullerenes which have a perfect star packing. Section $3$ gives
a negative answer to the problem “Is there a fullerene on $8n+4$ vertices with
a perfect star packing?” asked by Doslić et. al [13]. This implies that a
fullerene graph with an efficient dominating set must has $8n$ vertices. In
section $4$, we generalize the Proposition $1$ in reference [13] and give
three counterexamples for Theorem $14$ in the same paper. And some forbidden
configurations are listed to preclude the existence of a perfect star packing
of type $P0$.

##  2 Characterization of fullerenes with a perfect star packing

A _fullerene_ graph (simply fullerene) is a cubic $3$-connected plane graph
with only pentagonal and hexagonal faces. It follows from the Euler formula
that there must be exactly $12$ pentagons in every fullerene graph. Such
graphs are suitable models for carbon fullerene molecules: carbon atoms are
represented by vertices, whereas edges represent chemical bonds between two
atoms (see [15, 25]). In a classical paper by Grünbaum and Motzkin [18], we
know that a fullerene graph with $n$ vertices exists for all even $n\geq 20$
except for $n=22$. Klein and Liu [23] used a similar approach to show that
there exist fullerene graphs on $n$ vertices with isolated pentagons for
$n=60$ and for each even $n\geq 70$. We refer the reader to the monograph [15]
for a systematic introduction on fullerene graphs.

A cycle of a fullerene graph $G$ is a _facial cycle_ if it is the boundary of
a face in $G$, otherwise, it is a _non-facial_ cycle. Clearly, each pentagon
and hexagon in $G$ is a facial cycle since $G$ is $3$-connected and any
$3$-edge-cut is trivial [31]. In paper [13], the authors obtained the
following basic conclusions.

######  Proposition 2.1 ([13]).

Let $S$ be a perfect star packing of fullerene graph $G$. Then each pentagon
of $G$ can contain at most one center of a star in $S$.

######  Lemma 2.2 ([13]).

Let $S$ be a perfect star packing of fullerene graph $G$. Then a vertex shared
by two pentagons of $G$ cannot be the center of a star in $S$.

Recall that a vertex set $X$ of a graph $G$ is said to be _independent_ if any
two vertices in $X$ are not adjacent in $G$. A cycle $C=v_{1}v_{2}\cdots
v_{k}v_{1}$ in $G$ is called _induced_ if $v_{i}$ has only two adjacent
vertices $v_{i+1}$ and $v_{i-1}$ around the $k$ vertices
$v_{1},v_{2},\cdots,v_{k}$ (note that if $i=k,i+1:=1$ and if $i=1,i-1:=k$).
Otherwise, there exists some $i$ and $j\notin\\{i-1,i+1\\}$ such that $v_{i}$
and $v_{j}$ are adjacent in $G$, the edge $v_{i}v_{j}$ is a _chord_ of $C$ and
$C$ is not induced. A subgraph $R$ of a graph $G$ is _spanning_ if $R$ covers
all the vertices of $G$. For a vertex $v$ of a graph $G$, we call vertex $u$
being a _neighbor_ of $v$ in $G$ if $u$ is adjacent to $v$ in $G$.

######  Theorem 2.3.

Let $G$ be a fullerene graph. Then $G$ has a perfect star packing if and only
if $G$ has an independent vertex set $S^{*}$ such that each component of
$G-S^{*}$ is an induced cycle in $G$.

###### Proof.

If $G$ has a perfect star packing $S$, then $S$ is a spanning subgraph of $G$
and any component in $S$ is isomorphic to a star graph $K_{1,3}$. Let $S^{*}$
be the set of all $3$-degree vertices in $S$. Clearly, $S^{*}$ is an
independent vertex set in $G$ and any vertex in $G-S^{*}$ has degree $2$. So
each component of $G-S^{*}$ is an induced cycle in $G$.

Let $S^{*}$ be an independent vertex set of $G$ such that each component of
$G-S^{*}$ is an induced cycle in $G$. Clearly, each vertex in $S^{*}$ and its
three neighbors induce a star graph $K_{1,3}$. We collect all these star
graphs and denote this set by $\mathcal{H}$. For any vertex $x$ on a cycle $C$
in $G-S^{*}$, $x$ has exactly one neighbor in $S^{*}$ since $G$ is $3$-regular
and induced cycle $C$ is a component of $G-S^{*}$. So $\mathcal{H}$ is a
spanning subgraph of $G$ and each component of $\mathcal{H}$ is a star graph
$K_{1,3}$, that is, $\mathcal{H}$ is a perfect star packing of $G$. ∎

We note that star graph $K_{1,3}$ has exactly one _center_ (the vertex of
degree $3$) and three leaves. A perfect star packing $S$ of a fullerene graph
$G$ is a spanning subgraph of $G$ each component of which is a star graph
$K_{1,3}$. We call each $1$-degree vertex in $S$ being a _leaf_. In the
following, we denote by $C(S)$ the set of all the centers of stars in $S$.

######  Remark 2.4.

Let $S$ be a perfect star packing of fullerene graph $G$. Then
$1.$ $C(S)$ is an independent vertex set in $G$.
$2.$ Any leaf in $S$ has exactly one neighbor belonging to $C(S)$ and has
exactly two neighbors being leaves in $S$.
$3.$ Each cycle in $G-C(S)$ does not have a chord.

######  Proposition 2.5.

Each hexagon can contain at most two centers of a perfect star packing of
fullerene graph $G$. If a hexagon $h$ contains two such centers, then they are
antipodal points on the hexagon $h$.

###### Proof.

Let $h$ be a hexagon in $G$. We denote the six vertices of $h$ by
$v_{1},v_{2},\dots,v_{6}$ in the clockwise direction. If vertex $v_{1}$ is the
center of a star $H$ in a perfect star packing $S$ of $G$, then $v_{2}$ and
$v_{6}$ are two leaves in $H$. Hence both $v_{3}$ and $v_{5}$ are leaves in
$S$ by Remark 2.4 2\. Clearly, $v_{4}$ could be the center of a star in $S$.
Hence $h$ has exactly one center of $S$ or has exactly two centers of $S$
which are antipodal points on $h$. ∎

##  3 The order of fullerenes with a perfect star packing

To show the main conclusion, we need to prepare as follows.

Fig. 1: (a) Type $1$; (b) Type $2$; (c) Type $3$.

######  Lemma 3.1.

Let $S$ be a perfect star packing of fullerene graph $G$. Then for any vertex
$x\in C(S)$, all the vertices on the three faces sharing $x$ are covered by
$S$ as Type $1$, Type $2$ or Type $3$ _(_ see Fig. 1 _)_.

###### Proof.

By the Lemma 2.2, at most one of the three faces sharing $x$ is a pentagon
since $x\in C(S)$. There are two cases as follows.

Case 1. The three faces sharing $x$ are all hexagons.

Clearly, $x$ has three antipodal points on the three hexagons sharing $x$,
denoted by $x_{1}$, $x_{2}$ and $x_{3}$ respectively as depicted in Fig. 1
(a). By Remark 2.4 2, the two neighbors $v_{1}$ and $v_{3}$ of $v_{2}$ are
leaves in $S$. Similarly, $u_{1},u_{3},w_{1}$ and $w_{3}$ are also leaves in
$S$. We claim that at least two of $x_{1},x_{2}$ and $x_{3}$ are centers of
stars in $S$. If $x_{1}$ is not the center of a star in $S$, then $x_{1}$ is a
leaf in $S$. So the third neighbor of $v_{1}$, say $y_{1}$, is the center of a
star in $S$ (see Fig. 1 (b)). Similarly, the third neighbor of $w_{3}$, say
$y_{2}$, is also the center of a star in $S$. Since the three vertices
$v_{1},v_{2}$ and $v_{3}$ are leaves in $S$ and $y_{1}\in C(S)$, the face
$f_{1}$ has only one center of $S$ by Propositions 2.5 and 2.1. Hence the two
neighbors of $v_{3}$ on $f_{1}$ are leaves. By Remark 2.4 2, $x_{3}$ is the
center of a star in $S$, that is, $x_{3}\in C(S)$. Similarly, $w_{1}$ is a
leaf in $S$ and the two neighbors of $w_{1}$ on $f_{2}$ are all leaves in $S$.
Hence $x_{2}\in C(S)$. So at least two of $x_{1}$, $x_{2}$ and $x_{3}$ belong
to $C(S)$. If exactly two of $x_{1}$, $x_{2}$ and $x_{3}$ belong to $C(S)$,
without loss of generality, we suppose that $x_{2},x_{3}\in C(S)$, then all
the vertices on the three faces sharing $x$ are covered by $S$ as Type $2$. If
all the three vertices $x_{1}$, $x_{2}$ and $x_{3}$ belong to $C(S)$ (see Fig.
1 (a)), then all the vertices on the three faces sharing $x$ are covered by
$S$ as Type $1$.

Case 2. Exactly one of the three faces sharing $x$ is a pentagon.

By Proposition 2.1, $w_{1}$ and $u_{3}$ are leaves in $S$ (see Fig. 1 (c)).
Hence $x_{4},x_{3}\in C(S)$ and $f$ is a hexagon by Remark 2.4. 2 and
Proposition 2.5. By Remark 2.4. 2, the neighbor $w_{3}$ of $w_{2}$ is a leaf
in $S$ since the neighbor $x$ of $w_{2}$ belong to $C(S)$. Hence the other
vertices on $f_{1}$ except for $x_{4}$ are all leaves in $S$ by Propositions
2.1 and 2.5. This follows that the neighbor $x_{1}$ of $w_{3}$ is the center
of a star in $S$ by Remark 2.4 2\. Similarly, we can show $x_{2}\in C(S)$.
Hence all the vertices on the three faces sharing $x$ are covered by $S$ as
Type $3$ (see Fig. 1 (c)). ∎

######  Corollary 3.2.

Let $S$ be a perfect star packing of fullerene graph $G$. If a pentagon $P$ of
$G$ has a vertex $x\in C(S)$, then $G-C(S)$ has a non-facial cycle $C$ of $G$
such that the path $P-x$ is a subgraph of $C$.

###### Proof.

By Proposition 2.1, $x$ is shared by this pentagon $P$ and two hexagons. So
all the vertices on the three faces sharing $x$ are covered by $S$ as Type $3$
(see Fig. 1 (c)). Clearly, the path $P-x$ is a subgraph of a cycle $C$ in
$G-C(S)$ and $C$ is a non-facial cycle of $G$. ∎

We note that $3$-connected graphs have only one embedding up to equivalence
[11]. If we embed a fullerene graph $G$ in the plane, then any non-facial
cycle $C$ of $G$ as a Jordan curve separates the plane into two regions,
denoted by $R_{1}^{*}$ and $R_{2}^{*}$, each of which has the entire $C$ as
its frontier. We denote the subgraph of $G$ induced by the vertices lying in
the interior of $R_{i}^{*}$ by $G_{i}$, $i=1,2$. Here we note that
$\\{V(G_{1}),V(G_{2}),V(C)\\}$ is a partition of all the vertices of $G$. We
say that $C$ _divide_ the graph $G$ into two _sides_ $G_{1}$ and $G_{2}$.

######  Theorem 3.3.

Let $S$ be a perfect star packing of fullerene graph $G$ and $C$ be a cycle in
$G-C(S)$. Then $C(S)$ does not have a vertex which has three neighbors on $C$.

Fig. 2: $x\in C(S)$ has three neighbors on $C$.

###### Proof.

If $C$ is a facial cycle of $G$, then $C$ is a pentagon or a hexagon. The
conclusion clearly holds. Now, let $C$ be a non-facial cycle of $G$. Then $C$
divides $G$ into two sides, denoted by $H_{1}$ and $H_{2}$ respectively. We
note that all vertices on $C$ are leaves in $S$ since $C$ is a cycle in
$G-C(S)$. On the contrary, we suppose that there is a vertex $x\in C(S)$ which
has three neighbors on $C$, denoted by $x_{1},x_{2}$ and $x_{3}$ respectively.
Without loss of generality, we suppose that $x\in V(H_{1})$ (see Fig. 2 (a)).
The three vertices separate the circle $C$ into three sections, denoted by
$C_{1},C_{2}$ and $C_{3}$ respectively, each of which is a path with $x_{i}$
and $x_{i+1}$ as two terminal ends, $i=1,2,3$ (if $i=3$, then $i+1:=1$). From
Lemma 3.1 we know that at most one of $x_{1}C_{1}x_{2}x$, $x_{2}C_{2}x_{3}x$
and $x_{3}C_{3}x_{1}x$ is a facial cycle of $G$ since $C$ is a cycle in
$G-C(S)$. Next, we suppose that $x_{1}C_{1}x_{2}x$ and $x_{2}C_{2}x_{3}x$ are
non-facial cycles of $G$. Let $C_{1}=x_{1}v_{1}v_{2}\cdots v_{k}x_{2}$,
$C_{2}=x_{2}u_{1}u_{2}\cdots u_{t}x_{3}$. So $k\geq 5$ and $t\geq 5$ since any
non-facial cycle of $G$ has length at least $8$. By Remark 2.4.3, $C$ does not
have a chord. So $v_{1}v_{k}\notin E(G)$ and $u_{1}u_{t}\notin E(G)$. This
implies that $h$ is a hexagon face of $G$, and $x_{1},x,x_{2}$ and
$v_{1},v_{k}$ are five vertices on $h$. We denote the sixth vertex of $h$ by
$y$. Clearly, $y\in V(H_{1})$ by the planarity of $G$ (see Fig. 2 (b)).
Similarly, both $u_{1}$ and $u_{t}$ have a common neighbor in $H_{1}$.

Since $S$ is a perfect star packing of $G$ and the two neighbors $x_{1}$ and
$v_{2}$ of $v_{1}$ are leaves in $S$, $y$ is the center of a star in $S$. If
the third neighbor of $y$ is on $C$, then it is on $C_{1}$, denoted it by
$v_{r}$. The three neighbors of $y$ separate the circle $C$ into three
sections, two of which are subgraphs of $C_{1}$, denoted by $C_{1}^{1}$ and
$C_{1}^{2}$ respectively. As the above discussion, we know that one of
$v_{1}C_{1}^{1}v_{r}y$ and $v_{r}C_{1}^{2}v_{k}y$ is a non-facial cycle of
$G$. By the recursive process and the finiteness of the order of $G$, we can
suppose that the third neighbor of $y$ is not on $C$, and denoted it by
$y^{\prime}$.

See Fig. 2 (b), the five vertices $v_{k-1},v_{k},x_{2},u_{1},u_{2}$ belong to
a common facial cycle $h^{\prime}$ of $G$. Since $C$ does not have a chord by
Remark 2.4 3, $v_{k-1}$ and $u_{2}$ are not adjacent in $G$. So $h^{\prime}$
is a hexagon. By the planarity of $G$, $v_{k-1}$ and $u_{2}$ have a common
neighbor in $H_{2}$. so $v_{k-2},v_{k-1},v_{k},y$ and $y^{\prime}$ are on a
face of $G$, say $f$. If $f$ is a pentagon, then $v_{k-2}$ is adjacent to
$y^{\prime}$. So all the three neighbors of $v_{k-2}$ are leaves in $S$. This
implies a contradiction since $v_{k-2}$ is also a leaf in $S$. If $f$ is a
hexagon, then $v_{k-2}$ and $y^{\prime}$ have a common neighbor, denoted by
$z$. Clearly, $z$ is $v_{k-3}$ or not. For $z=v_{k-3}$, the three neighbors of
$v_{k-3}$ are all leaves in $S$, a contradiction. For $z\neq v_{k-3}$, by
Remark 2.4 2, $z$ is a leaf in $S$ since $y^{\prime}$ has a neighbor $y\in
C(S)$. So the three neighbors of $v_{k-2}$ are all leaves in $S$, a
contradiction. All these contradictions imply that $C(S)$ does not have a
vertex which has three neighbors on $C$. ∎

Let $S$ be a perfect star packing of fullerene graph $G$ and $C$ be a cycle in
$G-C(S)$ which is a non-facial cycle of $G$. $C$ divides $G$ into two sides,
denoted by $H_{1}$ and $H_{2}$ respectively. Set $C^{i}$ be the set of all the
vertices on $C$ each of which has a neighbor in $H_{i}$, $i=1,2$. Clearly,
$\\{C^{1},C^{2}\\}$ is a partition of $V(C)$. $G[C^{i}]$ is a vertex induced
subgraph of $G$ which has vertex set $C^{i}$ and any two vertices of $C^{i}$
are adjacent if and only if they are adjacent in $G$. See Fig. 4, $G[C^{1}]$
is depicted as red and $G[C^{2}]$ is depicted as blue. In the following, we
use these symbols no longer explaining.

######  Lemma 3.4.

For $i=1,2$, if a vertex $x$ on $C$ has a neighbor in $H_{i}$, then the
component of the induced subgraph $G[C^{i}]$ which contains $x$ is a path with
$2$ or $3$ vertices.

###### Proof.

We suppose that $x$ on $C$ has adjacent vertex in $H_{1}$. For the convenience
of the following description, set $C:=xv_{1}v_{2}\cdots v_{k}x$. Since $C$ is
a cycle in $G-C(S)$ which is a non-facial cycle of $G$, the length of $C$ is
at least $8$. So $k\geq 7$. There are three cases for the two neighbors
$v_{1}$ and $v_{k}$ of $x$ on $C$.

Case 1. Both $v_{1}$ and $v_{k}$ have neighbors in $H_{2}$.

In this case, the three vertices $v_{1},x$ and $v_{k}$ lie on the same face
$f$ of $G$ (see Fig. 3 (a)). Since all the vertices on $C$ are leaves in $S$,
the other neighbor of $v_{1}$ (resp. $v_{k}$) which is not on $C$ is the
center of a star in $S$. So $f$ has two vertices in $C(S)$ which are the
centers of two stars in $S$ covered $v_{1}$ and $v_{k}$, respectively. So $f$
is a hexagon by Proposition 2.1. But the case cannot hold by Propositions 2.5.

Fig. 3: (a) $v_{1}$ and $v_{k}$ have neighbors in $H_{2}$; (b) $v_{1}$ and
$v_{k}$ have neighbors in $H_{1}$; (c) $v_{1}$ has a neighbor in $H_{1}$,
$v_{k}$ has one in $H_{2}$.

Case 2. Both $v_{1}$ and $v_{k}$ have neighbors in $H_{1}$.

In this case, the five vertices $v_{2},v_{1},x,v_{k},v_{k-1}$ belong to a
facial cycle $h$ of $G$ (see Fig. 3 (b)). We claim that both $v_{2}$ and
$v_{k-1}$ have neighbors in $H_{2}$. Otherwise, at least one of $v_{2}$ and
$v_{k-1}$ has a neighbor in $H_{1}$. If $v_{2}$ has a neighbor in $H_{1}$ and
$v_{k-1}$ has a neighbor in $H_{2}$, then the six vertices
$v_{3},v_{2},v_{1},x,v_{k},v_{k-1}$ lie on a face $h$ of $G$. So $h$ is a
hexagon and $C$ has a chord $v_{3}v_{k-1}$, a contradiction. For $v_{2}$
having a neighbor in $H_{2}$ and $v_{k-1}$ having a neighbor in $H_{1}$, we
can also obtain a chord of $C$, a contradiction. If both $v_{2}$ and $v_{k-1}$
have neighbors in $H_{1}$, then the seven vertices
$v_{3},v_{2},v_{1},x,v_{k},v_{k-1},v_{k-2}$ belong to a common face $h$ of
$G$. This implies that $G$ has a facial cycle of length at least $7$, a
contradiction. So both $v_{2}$ and $v_{k-1}$ have neighbors in $H_{2}$, and
$v_{2},v_{1},x,v_{k},v_{k-1}$ lie on a hexagon $h$ of $G$ (see Fig. 3 (b)).
Since $C$ does not have a chord, the path $v_{1}xv_{k}$ is a connected
component of the induced subgraph $G[C^{1}]$.

Case 3. $v_{1}$ has a neighbor in $H_{1}$ and $v_{k}$ has a neighbor in
$H_{2}$, or $v_{1}$ has a neighbor in $H_{2}$ and $v_{k}$ has a neighbor in
$H_{1}$.

By the symmetry, it is sufficient to consider that $v_{1}$ has a neighbor in
$H_{1}$ and $v_{k}$ has a neighbor in $H_{2}$. If $v_{2}$ has a neighbor in
$H_{1}$, then $v_{3}$ must have a neighbor in $H_{2}$, otherwise, $C$ has a
chord or $G$ has a facial cycle of length at least seven, a contradiction. As
the proof of Case 2, $v_{3},v_{2},v_{1},x,v_{k}$ lie on a hexagonal facial
cycle. So the path $v_{2}v_{1}x$ is a connected component of the induced
subgraph $G[C^{1}]$. Now, we suppose that $v_{2}$ has a neighbor in $H_{2}$.
Then the four vertices $v_{k},x,v_{1},v_{2}$ lie on the same face $g$ of $G$.
Since $v_{k},x,v_{1},v_{2}$ are all leaves in $S$, $g$ is a pentagon and
$v_{2},v_{k}$ have a common neighbor in $H_{2}$ which is the center of a star
in $S$ (see Fig. 3 (c)). So the path $xv_{1}$ is a connected component of the
induced subgraph $G[C^{1}]$.

In summary, the component of the induced subgraph $G[C^{1}]$ which contains
$x$ is a path with $2$ or $3$ vertices since $C$ does not have a chord. ∎

In addition, we have the following Lemma.

######  Lemma 3.5.

Each component of $G[C^{i}]$ is a path with $2$ or $3$ vertices, $i=1,2$.

Fig. 4: (a) A cycle $C$ of length $25$; (b) A cycle $C$ of length $30$.
($G[C^{1}]$ is red and $G[C^{2}]$ is blue.)

###### Proof.

For any vertex $x$ on $C$, $x$ must have exactly one neighbor in $H_{1}$ or
$H_{2}$ since $G$ is $3$-regular and $C$ does not have a chord. Without loss
of generality, we suppose that $x$ has exactly one neighbor in $H_{1}$. By
Lemma 3.4, the component of the induced subgraph $G[C^{1}]$ which contains $x$
is a path with $2$ or $3$ vertices . We note that the choice of $x$ is
arbitrary. So the conclusion holds. ∎

######  Proposition 3.6.

Let $C=v_{0}v_{1}\cdots v_{k-1}$ be a non-facial cycle in $G-C(S)$. $($In the
following, the subscript is module $k$$)$.
$(i)$ If both $v_{i}$ and $v_{i+1}$ have neighbors in $H_{1}$ $($or $H_{2}$$)$
and $v_{i-1}$ and $v_{i+2}$ have neighbors in $H_{2}$ $($or $H_{1}$$)$, then
the four vertices $v_{i-1},v_{i},v_{i+1}$ and $v_{i+2}$ lie on a pentagon of
$G$.
$(ii)$ If $v_{i},v_{i+1},v_{i+2}$ have neighbors in $H_{1}$ $($or $H_{2}$$)$
and $v_{i-1}$ and $v_{i+3}$ have neighbors in $H_{2}$ $($or $H_{1}$$)$, then
the five vertices $v_{i-1},v_{i},v_{i+1},v_{i+2}$ and $v_{i+3}$ lie on a
hexagon of $G$.
$(iii)$ For $j=1,2$, if both $v_{i}$ and $v_{i+1}$ have neighbors in $H_{j}$
$($we denote the two edges incident to $v_{i}$ and $v_{i+1}$ not lie in $C$ by
$e_{i}$ and $e_{i+1}$, respectively$)$, then the facial cycle containing both
$e_{i}$ and $e_{i+1}$ is a hexagon, and two antipodal points on this hexagon
are centers of two stars in the perfect star packing $S$.

###### Proof.

Cases $(i)$ and $(ii)$ can be easily obtained from the proof of the Cases $2$
and $3$ of Lemma 3.4 (see Fig. 3). Since all the vertices on $C$ are leaves in
the perfect star packing $S$, the other end of $e_{i}$ (resp. $e_{i+1}$) which
is not on $C$, denoted by $u_{i}$ (resp. $u_{i+1}$), is the center of a star
in $S$. We know that any facial cycle of $G$ is a pentagon or a hexagon. So
$u_{i}$ and $u_{i+1}$ are distinct. By Lemmas 2.1 and 2.5, the facial cycle
containing both $e_{i}$ and $e_{i+1}$ is a hexagon, and $u_{i}$ and $u_{i+1}$
are antipodal points on this hexagon. ∎

For example, in Fig. 4, except for $f_{i},i\in\\{1,2,3,4,5\\}$ the other faces
sharing edges with $C$ are all hexagons. Moreover, how the vertices on $C$
being covered by $S$ is determined.

We recall that the the union of two graphs $G_{1}$ and $G_{2}$ is denoted by
$G_{1}\cup G_{2}$, which has vertex set $V(G_{1})\cup V(G_{2})$ and edge set
$E(G_{1})\cup E(G_{2})$. Let $n_{3}$ be the number of the components of
$G[C^{1}]\cup G[C^{2}]$ each of which is isomorphic to a path with $3$
vertices. Similarly, $n_{2}$ is the number of the components of $G[C^{1}]\cup
G[C^{2}]$ each of which is isomorphic to a path with $2$ vertices. For
example, $n_{3}=n_{2}=5$ in Fig. 4 (a) and $n_{3}=10$, $n_{2}=0$ in Fig. 4
(b).

######  Observation 1.

$n_{2}+n_{3}$ is even.

######  Proposition 3.7.

Let $S$ be a perfect star packing of fullerene graph $G$ and $C$ a cycle in
$G-C(S)$ which is a non-facial cycle of $G$. Then the length of $C$ is
$3n_{3}+2n_{2}$, and has the the same parity with $n_{2}$ and $n_{3}$.

###### Proof.

Clearly, the length of $C$ is $3n_{3}+2n_{2}$ by Lemma 3.5. So $n_{3}$ is odd
if and only if the length of $C$ is odd. Since $n_{2}+n_{3}$ is even by
Observation 1, the parity of $n_{2}$ and $n_{3}$ are same. Then we are done. ∎

######  Theorem 3.8.

Let $S$ be a perfect star packing of fullerene graph $G$. Then $G-C(S)$ has
even number of odd cycles.

###### Proof.

If $G-C(S)$ does not have a non-facial cycle of $G$, then any pentagon of $G$
does not have a vertex in $C(S)$ by Corollary 3.2. So all the vertices on
pentagons are leaves in $S$. It implies that $G-C(S)$ has exactly twelve odd
cycles, each of which is a pentagon. Next, we suppose that $G-C(S)$ has a non-
facial cycle of $G$, denoted by $C$.

Claim 1. If $C$ is an even cycle, then $G$ has even number of pentagons which
share edges with $C$. If $C$ is an odd cycle, then $G$ has odd number of
pentagons which share edges with $C$.

By Proposition 3.6, the number of pentagons which share edges with $C$ is
equal to $n_{2}$. By Proposition 3.7, $n_{2}$ and the length of $C$ have the
same parity. So the Claim holds.

Claim 2. Any pentagon of $G$ shares edges with at most one non-facial cycle in
$G-C(S)$.

Let $P$ be a pentagon of $G$. By Proposition 2.1, $P$ has at most one vertex
which is the center of a star in $S$. If $P$ does not have a vertex in $C(S)$,
then $P$ is a cycle in $G-C(S)$. By Theorem 2.3, each component of $G-C(S)$ is
an induced cycle of $G$. So $P$ does not share edges with any non-facial cycle
in $G-C(S)$. If $P$ has a vertex $x\in C(S)$, then by Corollary 3.2 $P-x$ is a
subgraph of a non-facial cycle in $G-C(S)$. So $P$ shares edges with exactly
one non-facial cycle in $G-C(S)$.

Now, we consider the following two cases for the non-facial cycles in
$G-C(S)$.

Case 1. $G-C(S)$ does not have a non-facial cycle of odd length.

Then any non-facial cycle $C$ in $G-C(S)$ is of even length. By the above
Claims, there are even number of pentagons in $G$ such that they share edges
with $C$. Since $G$ has exactly twelve pentagons, there are even number of
pentagons in $G$ each of which does not share edges with non-facial cycles in
$G-C(S)$. These pentagons must be cycles in $G-C(S)$ by Corollary 3.2. Hence
$G-C(S)$ has even number of odd cycles.

Case 2. $G-C(S)$ has some non-facial cycle of odd length.

Suppose that $G-C(S)$ has exactly $k$ non-facial cycles of odd length. We
denote the number of pentagons in $G$ each of which does not share edges with
non-facial cycles in $G-C(S)$ by $p$. These $p$ pentagons must be cycles in
$G-C(S)$ by Corollary 3.2. So $G-C(S)$ has $p+k$ odd length cycles. Next, we
show that $p$ and $k$ have the same parity. If $p$ is odd, then $G$ has odd
number of pentagons each of which share edges with exactly one non-facial
cycle in $G-C(S)$ since $G$ has exactly $12$ pentagons. By the above Claims,
for each even length non-facial cycle in $G-C(S)$, $G$ has even number of
pentagons which share edges with the cycle, and for each odd length non-facial
cycle in $G-C(S)$, $G$ has odd number of pentagons which share edges with the
cycle. So $G-C(S)$ has odd number of non-facial cycles of odd length. This
means that $k$ is odd. For $p$ being even, we can similarly show that $k$ is
even. So $k$ and $p$ have the same parity and $p+k$ is even. ∎

Clearly, for a fullerene graph $G$ with a perfect star packing, its order must
be divisible by $4$. So the order of $G$ is $8k$ or $8k+4$ for some positive
integer $k$. Now, we can obtain the following main theorem which illustrate
that the order of $G$ can not be $8k+4$.

######  Theorem 3.9.

If fullerene graph $G$ has a perfect star packing, then the order of $G$ is
divisible by $8$.

###### Proof.

We suppose that $S$ is a perfect star packing of $G$ and $\mathcal{C}_{o}$ and
$\mathcal{C}_{e}$ are the collections of all the odd cycles and even cycles in
$G-C(S)$, respectively. Then we have the following equation.

$\begin{split}|V(G)|&=|C(S)|+\sum_{C\in\mathcal{C}_{o}}|C|+\sum_{C\in\mathcal{C}_{e}}|C|\\\
&=\frac{|V(G)|}{4}+\sum_{C\in\mathcal{C}_{o}}|C|+even.\\\ \end{split}$ (1)

By Theorem 3.8, $\mathcal{C}_{o}$ has even number of elements. Combine the
above equation, we know that $\frac{|V(G)|}{4}\times 3$ is even. Hence
$\frac{|V(G)|}{4}$ is even, that is, the order of $G$ is divisible by $8$. ∎

This theorem is equivalent to the following corollary.

######  Corollary 3.10.

A fullerene graph with order $8n+4$ does not have a perfect star packing.

We recall that a _dominating set_ of a graph $G$ is a set of vertices $D$ such
that each vertex in $V(G)-D$ is adjacent to a vertex in $D$. Moreover, if each
vertex in $V(G)-D$ is adjacent to exactly one vertex in $D$ and $D$ is an
independent vertex set, then $D$ is called _efficient_. The problem of
determining the existence of efficient dominating sets in some families of
graphs was first investigated by Biggs [4] and Kratochvil [24]. Later
Livingston and Stout [26] studied the existence and construction of efficient
dominating sets in families of graphs arising from the interconnection
networks of parallel computers. The problem of finding an efficient dominating
set, however, is algorithmically hard [2]. For more results and some
historical background regarding efficient dominating set, we refer the reader
to [8, 9, 22, 10] etc.

From the definitions of the efficient dominating set and the perfect star
packing of a fullerene graph $G$, the following proposition is a natural
result.

######  Proposition 3.11.

A fullerene graph $G$ with $n$ vertices has a perfect star packing if and only
if $G$ has an efficient dominating set of cardinality $\frac{n}{4}$.

Combine Theorem 3.9 and Proposition 3.11, we get the following theorem.

######  Theorem 3.12.

The order of a fullerene graph with an efficient dominating set is $8n$.

##  4 Some other conclusions

Došlić et. al. gave the following spectral necessary condition for the
existence of a perfect star packing in a fullerene graph.

######  Proposition 4.1 ([13]).

If a fullerene graph $G$ has a perfect star packing, then $-1$ must be an
eigenvalue of the adjacency matrix of $G$.

The proof of this Theorem can be translate to a simple $r$-regular graph. Here
for completeness, we prove as follows. For the definition of eigenvalues of
the adjacency matrix of a graph, we refer the reader to [17].

######  Theorem 4.2.

If a simple $r$-regular graph $G$ has a perfect $K_{1,r}$-packing $S$, then
$-1$ must be an eigenvalue of the adjacency matrix of $G$.

###### Proof.

Let $C(S)$ be the set of centers of stars $K_{1,r}$ in $S$. We define the
characteristic vector $\overrightarrow{c}\in\mathbb{R}^{|V(G)|}$ of $C(S)$ as
follows: $c_{i}=1$ if $i\in C(S)$, otherwise $c_{i}=0$. Set
$\overrightarrow{u}$ be the vector of all ones. For the adjacency matrix $A$
of $G$, we have $A\overrightarrow{u}=r\overrightarrow{u}$ since $G$ is
$r$-regular. Let
$\overrightarrow{w}=\overrightarrow{u}-(r+1)\overrightarrow{c}$. As
$A\overrightarrow{c}=\overrightarrow{u}-\overrightarrow{c}$, we have

$A\overrightarrow{w}=A\overrightarrow{u}-(r+1)A\overrightarrow{c}=r\overrightarrow{u}-(r+1)\overrightarrow{u}+(r+1)\overrightarrow{c}=(r+1)\overrightarrow{c}-\overrightarrow{u}=-\overrightarrow{w}$
(2)

This implies that $-1$ is an eigenvalue of $A$. ∎

A perfect star packing $S$ of a fullerene graph $G$ is of _type $P0$_ if no
center of a star in $S$ is on a pentagon of $G$. For such perfect star
packing, the following corollary holds.

######  Corollary 4.3.

If fullerene graph $G$ has a perfect star packing $S$ of type $P0$, then
$G-C(S)$ does not have a non-facial cycle of odd length.

###### Proof.

By the contrary, we suppose that $G-C(S)$ has a non-facial cycle $C$ of odd
length. By the Claim $1$ of Theorem 3.8, $G$ has a pentagon $P$ which share
edges with $C$. This implies that $P$ contains the center of a star in $S$.
This contradicts that $S$ is of type $P0$. So $G-C(S)$ does not have a non-
facial cycle of odd length. ∎

In the above Corollary, we note that $G-C(S)$ may have non-facial cycles of
even lengths (see Fig. 5, the blue cycle in $C_{120}$).

Now, we point out the error of the Theorem $14$ in [13].

######  Theorem 4.4 ([13]).

A fullerene graph on $8n$ vertices has a perfect star packing of type $P0$ if
and only if it arises from some other fullerene via the chamfer
transformation.

Fig. 5: Each of $C_{120},C_{144},C_{384}$ has a unique perfect star packing of
type $P0$ which is depicted in bold edges.

In the proof of the necessity of this Theorem, there exist the following
problem. Take a fullerene graph with a perfect star packing of type $P0$. All
star centers lie on vertices shared by three hexagons. When we connect the
centers of stars lying on the same hexagons, the resulting graph is planar,
but does not have to be $3$-regular, $3$-connected and have only pentagonal
and hexagonal faces. For example, it is easy to check that each of the
fullerene graphs $C_{120},C_{144},C_{384}$ (see Fig. 5) has a unique perfect
star packing of type $P0$. When we connect the centers of stars lying on the
same hexagons, the resulting graph (the red dashed line in Fig. 5 is the
resulting graph for $C_{120}$, and here we omit the resulting graphs for
$C_{144}$ and $C_{384}$) is planar and is not connected. In fact, the three
fullerene graphs $C_{120},C_{144}$ and $C_{384}$ as depicted in Fig. 5 cannot
arise from some other fullerene via the chamfer transformation. So the
necessity of this theorem does not hold, however, its sufficiency is right. It
can be corrected as follows.

######  Theorem 4.5.

A fullerene graph that arises from some other fullerene via the chamfer
transformation must have a perfect star packing of type $P0$.

From paper [13], we know that fullerenes with two pentagons sharing an edge
can not have a perfect star packing of type $P0$ since the edge shared by the
two pentagons cannot lie in any star. Next we list some other forbidden
configurations whose presence in a fullerene graph precludes the existence of
a perfect star packing of type $P0$.

Fig. 6: Three forbidden configurations.

######  Proposition 4.6.

If a fullerene graph $G$ contains a subgraph $PP1,PP3$ or $PP4$ _(_ see Fig. 6
_)_ , then it cannot have a perfect star packing of type $P0$.

###### Proof.

By the contrary, we suppose that $G$ has a perfect star packing of type $P0$,
denoted by $S$. Clearly, the vertices $v_{1}$ and $v_{2}$ (see Fig. 6) are
leaves in $S$. If $PP4$ is a subgraph of $G$, then $x_{1}$ is the center of a
star in $S$ since all vertices on a pentagon are leaves in $S$. So $x_{2}$ is
a leaf in $S$. By Remark 2.4 2, the neighbor $x_{3}$ of $x_{2}$ is also a leaf
in $S$. This implies that all the three neighbors of $v_{2}$ are leaves in
$S$, a contradiction. For subgraphs $PP1$ and $PP3$, we can similarly show
that $v_{2}$ has all its three neighbors being leaves in $S$, a contradiction.
∎

##  5 Acknowledgments

I would like to sincerely thank Wuyang Sun for his careful reading and
valuable comments and suggestions.

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c-number rather then an operator. The rescaled generators
$\mathsf{B}=\frac{B}{\sqrt{1-c^{2}}}$, $\mathsf{E}=E$,
$\mathsf{P}=\frac{P}{\sqrt{1-c^{2}}}$ then generate an
$\mathfrak{sl}_{2}(\mathbb{R})$ algebra. The 1-matter chord representation
described above transforms under this algebra, implying its action on two-
point functions [25]. This turns out to be the natural action of
$\mathfrak{sl}_{2}(\mathbb{R})$ on a circle with the circumference
$\beta_{\text{fake}}$, the fake circle described in Section 3. Moreover,
around small perturbations to the thermofield double state
$2c\mathsf{B}=T_{R}-T_{L}$ as an operatorial equation, implying an exponential
growth in Lorentzian time for the operator
$\mathsf{P}^{-}=\mathsf{E}-\mathsf{P}$ (which we will treat as the
“scramblon”),

$e^{i(T_{R}-T_{L})t}\mathsf{P}^{-}e^{-i(T_{R}-T_{L})t}=e^{2ct\mathbb{J}}\mathsf{P}^{-}\,,$
(5.27)

and suggesting that the Lypaunov exponent is
$\lambda_{L}=2c=2\pi/\beta_{\text{fake}}$ [25] as in Section 3.

##  6 Generalizations and variations

We have seen in Section 2 that the chord diagram technique can be applied to
several $p$-local systems in the double scaling limit. Actually, from the
discussion there it is clear that it applies to a large class of models.
Additional models analyzed so far include the double scaled sparse-SYK [66],
the commuting SYK of [67] (whose double scaling limit was studied in [59,
68]), models of interacting Paulis such as [69] and the more general polarized
spin operators of [68]. Here we are going to highlight additional models which
demonstrate some peculiar features, relevant for holography or for quantum
chaos.

###  6.1 The Parisi hypercube model

The first example is the Parisi hypercube model, introduced and solved in [70]
and recently studied in [71]. It is a case where chord diagrams are applied to
random Hamiltonians that go beyond $p$-locality (a partial characterization of
this class was suggested in [72, 73]), and presents a new take on the role of
frustration in chord models. One can use this approach of frustration to
construct models with bosons [74], which are otherwise difficult to control.

In the conventions of [72, 73], the hypercube has $2^{d}$ vertices denoted by
$\\{-\tfrac{1}{2},+\tfrac{1}{2}\\}^{d}$ with $\frac{\sigma^{3}_{\mu}}{2}$
being the position operators of the particle, $\mu=1,\cdots,d$. The
Hamiltonian is

$H=-\frac{1}{\sqrt{d}}\sum_{\mu=1}^{d}D_{\mu}\,,\quad
D_{\mu}=T_{\mu}^{+}+T_{\mu}^{-},\quad
T^{+}_{\mu}=\prod_{\nu=1,\nu\not=\mu}^{d}e^{\frac{i}{4}F_{\mu\nu}\sigma^{3}_{\nu}}\sigma^{+}_{\mu}\,,\quad
q\equiv\langle\cos F_{\mu\nu}\rangle\,,$ (6.1)

where $T^{-}_{\mu}=(T^{+}_{\mu})^{\dagger}$ and
$\sigma^{+}_{\mu}=\frac{\sigma^{1}_{\mu}+i\sigma^{2}_{\mu}}{2}$.
$\sigma_{\mu}^{i}$ ($i=1,2,3$) is the $i$-th Pauli matrix acting on the
$\mu$-th qubit, and $F_{\mu\nu}$ is the antisymmetric tensor of the background
flux. $T^{+}_{\mu}$ is a hopping operator that transports the particle in the
forward $\mu$ direction, while assigning to it a random phase due to a
disordered flux. We have chosen a normalization for $H$ such that it has a
compact spectral support at $d=\infty$ (as in DS-SYK). The fluxes $F_{\mu\nu}$
are quench-disordered, and identically and independently distributed with an
even distribution, but otherwise completely general. $\langle\cdots\rangle$
Denotes ensemble average, and $q$ is a tunable parameter.

The holonomy around a plaquette in the $\mu\nu$ plane,
$T^{-}_{\nu}T^{-}_{\mu}T^{+}_{\nu}T^{+}_{\mu}$, gives the return amplitude of
hopping counter-clockwise around it. It is more convenient to study the
holonomy in terms of the $D_{\mu}$ operator, which combines the forward and
backward hoppings:

${\cal
W}_{\mu\nu}=D_{\nu}D_{\mu}D_{\nu}D_{\mu}=\cos(F_{\mu\nu})-i\sin(F_{\mu\nu})\sigma^{3}_{\mu}\sigma^{3}_{\nu},\qquad\langle{\cal
W}_{\mu\nu}\rangle=q.$ (6.2)

We can also think of ${\cal W}_{\mu\nu}$ as the mutual frustration of
different terms in the Hamiltonian.

Consider now computing the moments
$\langle\operatorname{Tr}(H^{2k})\rangle=d^{-k}\sum_{\\{\mu_{i}\\}}\langle\operatorname{Tr}(D_{\mu_{1}}D_{\mu_{2}}\cdots
D_{\mu_{2k}})\rangle$ in the position basis. In the large $d$ limit, if a
direction $\mu$ appears in the trace, then it appears twice. This gives us the
basic chord diagram structure. If two chords cross it means that these four
hopping operators appear in the ordering (6.2), i.e. a chord intersection
gives a factor of $q$, and we obtain the chord rules right away.

###  6.2 Models with additional symmetries

There are some classic generalizations of the SYK models (actually, some
predate it) which are useful for experiments (the $U(1)$ invariant model) or
for holography (the SUSY models). From the chord perspective they are
interesting because the basic chord Hilbert space is seemingly a multi-chord,
with seemingly a very different set of intersection rules, yet it can be
reduced to a single chord Hilbert space. Beyond their own sake, it is an
interesting problem to explore the relation between theories that can be
reduced a simple transfer matrix and requirements of locality in the bulk.

The $U(1)$ invariant SYK model [75, 76, 77], was analyzed in the double
scaling limit in [78]. The model contains $N$ Dirac fermions, $\psi^{i}$, and
their complex conjugates ${\bar{\psi}}_{i}$

$H=\sum_{I,K}J_{I}^{K}{\bar{\psi}}_{K}\psi^{I}\,,$ (6.3)

where $I,K$ are multi-indices (whose length we double scale). The $J$’s are
complex Gaussian variable $J_{I}^{K}=(J_{K}^{I})^{*}$, and as before their
Wick contraction gives rise to a chords description. The weights of the chord
diagrams appear differently – the intersection of chords now has weight 1, and
non intersecting chords carry non-trivial weights that depend on the chemical
potential. Naively it seems that we cannot apply the transfer matrix
technique. However with some effort one can reduce them to the same transfer
matrix as above in a fixed charge sector. It is generally not clear what are
the total allowed set of pairwise chord interactions and which ones can
reduced to an ordinary transfer matrix.

The situation is more interesting in the supersymmetric SYK model [79] which
was solved in the double scaling in [65]. The model is

$Q\propto\sum_{I}C_{I}\Psi_{I},\ \ \ \ {\cal N}=1:\ C,\psi\ real;\ \ {\cal
N}=2:\ C,\psi\ complex$ (6.4)

The model with ${\cal N}=1$ is solved in terms of chords stretching between
insertions of $Q$. This means that $H_{effective}=T^{2}(q_{f})$ where $q_{f}$
is negative $q_{f}=-e^{-2p^{2}/N}$, and the Arik-Coon algebra (2.15) is then a
deformation of anti-commutation relations. The case of ${\cal N}=2$ is more
interesting. Since $Q^{2}=(Q^{\dagger})^{2}=0$, it is enough to consider
$\operatorname{Tr}(QQ^{\dagger}\cdots QQ^{\dagger})$. At each stage either $Q$
or $Q^{\dagger}$ can annihilate or create a chord. Since the $C$’s are complex
variables, the chords have an orientation and we will take it to go from a $Q$
to a $Q^{\dagger}$. When we cut the trace to go the transfer matrix we need to
keep track of whether a chord goes left or goes right, so there are two
species of chords, making the transfer matrix considerably more complicated,
but one can reduce it to the ordinary transfer matrix via decoupling a vast
number of null states in the chord Hilbert space.

This model was used in [80] to relate the fraction of ground states in the
full Hilbert space to the wavefunctions on the chord Hilbert space. In the
appropriate limit, it reproduces the Schwarzian result, while showing a series
of non-perturbative corrections to it whose interpretation is not yet
understood.

###  6.3 Multi chord problems

The generalization to multi-chord situation, i.e. when there are several types
of chords with non-trivial intersection rules is quite ubiquitous in problems
with RG flow. For example if we take $K$ independent random $p$-local
Hamiltonians on the same set of qubits with the total Hamiltonian being
$H=\sum_{i=1}^{K}\kappa_{i}H_{i}$, where $\kappa_{i}$ are the relative sizes
of the couplings. Each Hamiltonian has its own species of chords associated
with it, originating from the Wick contraction of its couplings. We assign a
weight $0\leq q_{ij}\leq 1$ [68] to the intersection of chords of types $i$
and $j$ (including $i=j$).

One can again define a chord Hilbert space and transfer matrix by introducing
$q$-deformed chord creation and annihilation operators [21, 64] and a transfer
matrix,

$[a_{i},a_{j}^{\dagger}]_{q_{ij}}=\delta_{ij}\,,\qquad
T=\sum_{i=1}^{K}\kappa_{i}\left(a_{i}+a_{i}^{\dagger}\right)\,,\qquad
Z(\beta)=\langle\vec{0}|e^{-\beta T}|\vec{0}\rangle\,.$ (6.5)

Unfortunately, if one tries to use the transfer matrix method then one needs
to track not just the total number of chords, but rather the number of chords
for each of the species as well as their ordering, making the Hilbert space
exponentially more complicated and rendering it impractical to work with (as
of yet) – even the size of the sub-space with $n$ chords in total is $K^{n}$.
Very little is known about the transfer matrix in this challenging case.

Nevertheless, one can write a “path intergal” formula for chords, which
simplifies considerably in the semi-classical limit $q_{ij}\to 1$ [68, 26]. At
the core of the approach one splits the thermal circle into small intervals
and uses variables that count how many chords of different types go between
all pairs of intervals – i.e. the density of chords between two points on the
thermal circle, $n_{i}(\tau_{1},\tau_{2})$. The expression282828The expression
in terms of the chord density only agrees with this one on-shell, and in
fluctuations around the saddle point. This issue has not been fully resolved
yet. looks especially suggestive when expressed in terms of the
variables292929When $K=1$ the variable $g(\tau_{1},\tau_{2})$ is proportional
to length of bulk geodesics that connect these two boundary points, and
therefore
$n(\tau_{1},\tau_{2})\sim\partial_{1}\partial_{2}g(\tau_{1},\tau_{2})$
connects the chord density to the Crofton form – the natural measure on the
space of bulk geodesics, see [81].,
$g_{i}(\tau_{1},\tau_{2})\sim\int_{\tau_{1}}^{\tau_{2}}d\tau_{3}\int_{\tau_{2}}^{\tau_{1}}d\tau_{4}\,n_{i}(\tau_{3},\tau_{4})$,
which counts how many chords go across a probe chord that would stretch
between the points $\tau_{1}$ and $\tau_{2}$,

$Z=\int\left(\prod_{i=1}^{K}\mathcal{D}g_{i}\right)\exp\left[-\frac{1}{\lambda}\int_{0}^{\beta}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\left[\sum_{i,j=1}^{K}\frac{1}{4}\alpha_{ij}g_{i}\partial_{1}\partial_{2}g_{j}+\sum_{i=1}^{K}\kappa_{i}^{2}\mathbb{J}^{2}e^{\sum_{j}\alpha_{ij}g_{j}}\right]\right]\,,$
(6.6)

where $q_{ij}=e^{-\lambda_{ij}}$, $\alpha_{ij}=\lambda_{ij}/\lambda$, and
$\lambda$ is some parameter that uniformizes the $q_{ij}\to 1$ limit. This
action303030We thank Josef Seitz and Ronny Frumkin for discussions of this
point. generalizes the $G\Sigma$ form (2.10) for these multi-chord diagrams.

A particularly interesting case [68, 26] arises when $K=2$ and the Hamiltonian
of the system has a chaotic part and an integrable part313131Such double
scaled Hamiltonians where $q=1$ were recently studied in [82].,

$H=\kappa_{1}H_{chaotic}+\kappa_{2}H_{integ}\,.$ (6.7)

For example, $H_{chaotic}$ could be the $p$-spin Hamiltonian (2.30) or the SYK
Hamiltonian (2.2) and $H_{integ}$ could be
$\sum_{I}J_{I}\sigma^{z}_{i_{1}}...\sigma^{z}_{i_{p^{\prime}}}$ or the
“commuting SYK” model [67], respectively. In these cases $q_{11}$ and $q_{12}$
are arbitrary ($<1$) and $q_{22}=1$. Using the above technique one can map the
phase diagram of the chaos to integrability transition and establish that it
is first order at low temperatures (perhaps with implications for the
experimental setup [83]).

Other Hamiltonians were also analyzed [68] – a mixture of SYK Hamiltonians
does not exhibit a phase transition [84], while other choices of $q_{ij}$’s
give a zero temperature phase transition.

##  7 Some additional frontiers

Finally, we’re going to briefly scan through some current topics of research
(in addition to some mentioned in the sections before), restricted to a level
of very rough pointers.

###  7.1 The bulk

We have seen a few time the appearance of unusual ”bulks”. One of them had to
do with Non-commutative geometry which is playing an important role in this
model, as it keeps showing up, in different guises, at almost every step. At
the same time, it is evident from Section 5 that there is no complete story
yet. The maximal goal would be a constructive derivation of the full 2D non-
commutative geometry (or integral over matrices, or some generalization of the
latter) directly from the microscopics, without any answer analysis.

The semi-classical fake geometries are another set of puzzling geometries
which emerges from the discussion. At face value one excises pieces of the
geometry for no apparent reason, and it is not clear (at least to us) how to
use them for higher $n$-pt functions. It might have to do with the fact the
inner product spreads across a finite volume of spacetime rather then being
localized on a Cauchy surface.

Some other issues which we have not discussed include:

####  7.1.0.0.1 A gravitational path integral

A natural question one might ask is whether the DS-SYK model can be recast in
the form of a gravitational path integral, where one has to integrate over
some bulk metrics of usual (commuting) spacetimes. Such a path integral
formulation was suggested in [85, 86, 56] and recently analyzed in [31]. It
takes the form of the sine-dilaton theory

$\int\mathcal{D}g\mathcal{D}\Phi\exp\left[\frac{1}{2\lambda}\int_{\mathcal{M}}d^{2}x\sqrt{g}\left(\Phi
R+2\sin(\Phi)\right)+\frac{1}{\lambda}\int_{\partial\mathcal{M}}d\tau\sqrt{h}\left(\Phi
K-e^{-\frac{i\Phi}{2}}\right)\right]\,,$ (7.1)

supplemented by the boundary conditions
$\Phi\big{|}_{\partial\mathcal{M}}=\frac{\pi}{2}+i\infty$,
$\sqrt{h}e^{-\frac{i\Phi}{2}}\big{|}_{\partial\mathcal{M}}=\frac{i}{\lambda}$.
The theory was canonically quantized in [31], analogously to JT-gravity [34],
by observing that a natural canonical coordinate is $L=\int
dse^{-\frac{i\Phi}{2}}$, the worldline action of a particle with non-minimal
coupling to the dilaton. The Hamiltonian of this system then becomes the
transfer matrix (3.3) as a finite difference operator, but additional steps
are needed to get the correct chord Hilbert space. When $\lambda\to 0$, these
steps include inserting a superposition of defects in the bulk, somewhat
resembling the fake disk picture. The resolution for finite $\lambda$ is
unclear.

This proposal also suggests a first order formulation (which generalizes JT’s
BF gauge theory formulation) in terms of a Poisson-sigma model [56, 53] – a
non-linear sigma model whose target space is the phase space of a quantum
mechanical system – which manifests the quantum group symmetry, and implies
the particle on a group description of Section 5.

####  7.1.0.0.2 Yang-Baxter moves and geometry

The relation between the chord picture and the bulk is confusing at a more
fundamental level. In the limit $q\rightarrow 1$ chord crossings are not
suppressed and they fill the disk, Figure 10(a). So a 2D bulk seems to arise
naturally, and we would like to have some sort of a map from chord diagrams to
metrics (for example).

(a)

(b)

Figure 10: (a) Chord diagrams proliferating at $q\rightarrow 1$, (b) The same
chord diagram but different bulk realizations.

But making this precise is tricky. Chord diagrams only provide information on
which chords intersect, and no information on how they traverse the bulk. In
fact we can move them around by “Yang-Baxter moves”. For example in 10(b) the
black chord has the same intersection numbers with the blue chords but it runs
in a different location relative to the blue mutual intersection. If we
believe that how chords traverse the bulk of the disk teaches us about the
bulk geometry, then these could be seen as different bulk geometries. So even
though 10(a) is suggestive, we cannot yet construct the bulk directly.

The main reason for obsessing about this is that a key paradigm in string
theory and field theory, due to ‘t Hooft, is that 2D surfaces are intimately
related to the double line large $N$ expansion. In the language of chords
these would be non-intersecting chords but with higher order vertices
(determined by the model’s potential). Here we have a different323232There
were attempts at reproducing the density of states and correlators of DS-SYK
using a 2-matrix model [87] (see also [88, 89]). For example, formally one can
take a multi-matrix ensemble and integrate it over orbits of the unitary
group. However, DS-SYK models are significantly different from ordinary RMT
ensembles in other features such as multi-trace correlators where they have
much larger early time connected contributions. way of obtaining 2D surfaces.

###  7.2 level statistics, multi trace correlators and wormholes

A key observable in any random systems is the Spectral Form Factor (SFF)
$\langle\operatorname{Tr}(e^{-(\beta+it)H})\operatorname{Tr}(e^{-(\beta-
it)H})\rangle_{J}$, which has in principle the same information as 2-energy
level statistics. For a Wigner-Dyson ensemble, the SFF is made out of a ramp
and a plateau (for a discussion in the context of SYK see [12, 16] and [90,
91] for JT gravity). This observable is just the simplest out all possible
multi trace correlators $\langle Z(\beta_{1})\cdots Z(\beta_{k})\rangle_{J}$.
In holography these are related to multiboundary Euclicdean wormholes.

However, in the full SYK model, as in any realistic model beyond the strict
Wigner-Dyson ensembles, there are much larger model dependent contributions at
shorter time scales, and to a large extent they are terra incognita.
(Un)Fortunately these are the time scales where the OTOC, Hawking radiation
and other interesting effects take place. These contributions were evaluated
in SYK or DS-SYK (the results are similar, but may differ for other chord
models) – their size in [92] and their exact form in [93]. These effects scale
as $1/N^{p}$ – much larger than the ramp and the plateau, which are
exponentially small in $N$. They correspond to global fluctuations of the
spectrum which shift its entire shape, and not just nearby energies (for the
implications to edge fluctuation see [94]).

Practically, this result is also relevant for sample-to-sample observable
quantities in potential experimental realization of the SYK system [95]. From
the point of view of gravity this is puzzling – for example, two boundaries of
size $\beta_{1}$ and $\beta_{2}$ “communicate” with each other via effects
which are much larger than what the “double trumpet” geometry predicts. It is
an open problem how to fully understand them in gravity [93].

###  7.3 Relation to (operator) complexity

A useful way of characterizing quantum chaos is by operator growth [96, 45].
There are several possibilities how to parameterize this growth, with Kyrlov
complexity being one popular method [45, 97]. The starting operator can also
be different. One option is to start with the identity operator and evolve it
by $e^{-{(\beta+it)}H}$ – this will measure333333also, it describes how the
complexity of the identity operator increases as we project it to subspaces of
lower energy using $e^{-\beta H}$. the complexity of the thermofield double
state343434Regarding the space of operators as the doubled Hilbert space., and
correspondingly the two sided wormhole, or its (upward) time evolution on both
sides. Another starting point could be a simple enough operator, and then we
evolve the wormhole with a single particle in it.

In the case of SYK the growth is parameterized in terms of the number of
fermions in the operators, i.e., by the super-operator
$\hat{n}(\mathcal{O})=\frac{1}{4p}\sum_{k=1}^{\infty}[\psi_{k},[\psi_{k},\mathcal{O}]]$
where $\mathcal{O}$ is the operator whose length we want to measure [98]. In
the case of DS-SYK it is just the number of chords: if we have $\hat{n}$
chords open, then the number of fermions in the state is ${\hat{n}}p$.
Microscopically, there are some fermions in the overlap of the index sets of
chords but they are of order ${\hat{n}}^{2}\ll{\hat{n}}p$ in the time regime
where the chord method applies. For the TFD state, the chord number measures
the Krylov state complexity [24]. We’ve seen before that $\hat{n}$ is also the
length of the wormhole connecting the two sides of the TFD state, and
therefore it provides a concrete realization of the idea that complexity is
encoded in the bulk geometry ([99, 100, 101] and many others). The length also
grows for simple operators (5.24), and for other models which have a chord
description. Specifically for the case of DS-SYK one can actually go further
and map the operator growth into motion in a Parisi hypercube model [72, 73].

###  7.4 de Sitter space

Some features of DS-SYK had prompted conjectures that at high temperatures it
describes the physics of quantum gravity on de Sitter space [102, 103, 42,
104, 105, 44, 106, 41, 107, 30, 108, 85].

First, its spectrum is bounded, as we have seen in Section 2. As such, the
(one-sided) Hamiltonians do not diverge in the large $N$ limit. In addition,
there is a maximally mixed state with finite energy, and equivalently, a
maximally entangled state on the doubled Hilbert space with finite energy,
which is the infinite temperature TFD state. In a related fashion, the
Hamiltonian and matter operators of DS-SYK generate a type II1 von-Neumann
operator algebra353535The connection between the $q$-oscillator algebra (2.15)
and type II1 operator algebras was long known in the mathematics community,
for example [23, 109]. [42], see also [62]. This is exactly the type of
algebra of observables in the static patch of de Sitter space [110].

Second, despite being at infinite temperature, there are still some notions of
finite temperature in the system. One is the fake temperature of Section 3,
that controls the decay of 2-pt functions. Another is the tomperature [42],
that measures the change in the energy of the system as the number of fermions
changes. Both have analogs in $dS_{2}$ [41], prompting a proposed duality
between the $\lambda\to 0$, infinite temperature limit of DS-SYK and JT
$dS_{2}$ [104, 105]. These proposals rely on the semi-classical features of
the models. More holographic properties at high temperatures were investigated
in [82].

Another proposal relates DS-SYK to pure gravity in three-dimensional de-Sitter
space, $dS_{3}$ [85, 30, 108, 107]. Interpreting two copies of the DS-SYK
model (with a constraint) as the worldlines of particles in $dS_{3}$, one can
match between correlation functions in the former and (S-wave) scalar Green’s
functions in the latter in the high-temperature, semi-classical limit [107].
But the proposal also extends to the quantum regime. The phase-space of
Schwarzschild-$dS_{3}$ solutions for pure gravity can be quantized. The
Hamiltonian is a certain line operator in the theory, which was shown to have
the same form as the DS-SYK transfer matrix (3.3) [108, 30], where $\lambda$
corresponds to the ratio between the Planck scale and the de Sitter radius.

Another evidence for the relation comes from treating pure gravity on dS3 as
an $SL(2,\mathbb{C})$ Chern-Simons theory, so the quantization described above
is a proposed quantization of the latter. The wavefunctions of states in its
Hilbert space are related to correlation functions in a system of two 2d
Liouville CFTs with conjugate central charges, $c_{\pm}=13\pm 12\pi
i(\frac{\lambda}{4\pi^{2}}-\frac{1}{\lambda})$, termed the the dS-Liouville
model. The boundary two point functions in this dS-Liouville model were then
shown to agree with the two-point functions of the doubled DS-SYK model
mentioned above for any $\lambda$ [30, 108].

It would be interesting to have a clearer picture of the relation between the
proposals, and to understand the role of the sine-dilaton gravity, given its
relation to the Liouville theory mentioned above, and the existence of
classical saddles in the theory where some regions in the bulk have positive
curvature [56].

####  7.4.0.0.1 Acknowledgments

We would like to thank A. Almheiri, A. Altland, A. Blommaert, N. Brukner, R.
Frumkin, F. Haehl, M. Isachenkov, Y. Jia, P. Narayan, V. Narovlansky, A. Raz,
S. Ross, M. Rozali, Josef Seitz, J. Simon and M. Watababe for many useful
discussions. OM is supported by the ERC-COG grant NP-QFT No. 864583 “Non-
perturbative dynamics of quantum fields: from new deconfined phases of matter
to quantum black holes”, by the MUR-FARE2020 grant No. R20E8NR3HX “The
Emergence of Quantum Gravity from Strong Coupling Dynamics”, and by the INFN
“Iniziativa Specifica GAST”. The work of MB is supported in part by the Israel
Science Foundation grant no. 2159/22, by the Minerva foundation, and by a
German-Israeli Project Cooperation (DIP) grant ”Holography and the Swampland”.

##  Appendix A Some $q$-functions

Here we define the special functions used throughout this work. We will use
$q=e^{-\lambda}$.

####  A.0.0.0.1 The $q$-Pochhammer symbol

The $q$-Pochhammer symbol is defined to be

$\left(z;q\right)_{n}=\prod_{k=0}^{n-1}\left(1-zq^{k}\right)\,,$ (A.1)

and the shorthand notation
$(a_{1},\cdots,a_{k};q)_{n}=\prod_{i=1}^{k}(a_{i};q)_{n}$ and $(e^{\pm
i\theta};q)_{n}=(e^{i\theta};q)_{n}(e^{-i\theta};q)_{n}$.

The infinite $q$-Pochhammer symbol has the expansions

$\left(z;q\right)_{\infty}=\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}q^{\frac{n\left(n-1\right)}{2}}}{\left(q;q\right)_{n}}z^{n}=\exp\left[-\sum_{k=1}^{\infty}\frac{1}{k}\frac{z^{k}}{1-q^{k}}\right]\,.$
(A.2)

####  A.0.0.0.2 The Jacobi theta function

The Jacobi theta functions are quasi-elliptic functions of two variables. We
will use the conventions of [111],

$\theta_{1}\left(u|\tau\right)=-i\sum_{k=-\infty}^{\infty}(-1)^{k}\mathfrak{q}^{(k+\frac{1}{2})^{2}}e^{\pi
i(2k+1)u}=2\mathfrak{q}^{1/4}\sin\left(\pi
u\right)\left(\mathfrak{q}^{2},\mathfrak{q}^{2}e^{\pm 2\pi
iu};\mathfrak{q}^{2}\right)_{\infty}\,,$ (A.3)

where $\mathfrak{q}=e^{\pi i\tau}$. The theta function has the modular
property

$\theta_{1}\left(u|\tau\right)=\frac{i}{\sqrt{-i\tau}}e^{-\pi
iu^{2}/\tau}\theta_{1}\left(\frac{u}{\tau}\,\bigg{|}-\frac{1}{\tau}\right)\,.$
(A.4)

####  A.0.0.0.3 $q$-numbers

We will sometimes use the definition of a $q$-number,

$[n]_{q}\equiv\frac{1-q^{n}}{1-q}=1+q+\cdots+q^{n-1}\,,$ (A.5)

where the last equality is valid for integers. When $q\to 1$ and $n$ is fixed,
$[n]_{q}\to n$.

####  A.0.0.0.4 The $q$-factorial and $q$-Gamma function

The $q$-Pochhammer symbol can also be used to define the $q$-factorial and
$q$-Gamma function,

$\Gamma_{q}\left(n\right)\equiv\left[n-1\right]_{q}!\equiv\prod_{k=1}^{n-1}[n]_{q}=\frac{\left(q;q\right)_{\infty}}{\left(q^{n};q\right)_{\infty}}\frac{1}{\left(1-q\right)^{n-1}}=\frac{(q,q)_{n-1}}{(1-q)^{n-1}}\,.$
(A.6)

In the limit $q\to 1$ the $q$-Gamma function becomes the usual Gamma function,
$\lim_{q\to 1}\Gamma_{q}(x)=\Gamma(x)$ when $x$ is independent of $q$.

####  A.0.0.0.5 The $q$-exponential

The $q$-deformation of the exponential function is363636In another standard
convention where
$e_{a}(z;\hat{q}^{2})\colonequals\sum_{n=0}^{\infty}{z^{n}\over\hat{q}^{an(n-1)\over
2}(\hat{q}^{2};\hat{q}^{2})_{n}}$ it is $e_{0}$.

$e_{q}(z)=e_{q}^{z}=\sum_{n=0}^{\infty}\frac{z^{n}}{[n]_{q}!}=\sum_{n=0}^{\infty}\frac{z^{n}(1-q)^{n}}{(q;q)_{n}}\,,\qquad
e_{q}\left(\frac{z}{1-q}\right)\xrightarrow{q\to 1}e^{z}\,.$ (A.7)

It has a nice property with respect to the $q$-derivative,
$D_{q}e_{q}^{z}={e_{q}(qz)-e_{q}(z)\over qz-z}=e_{q}(z)$.

####  A.0.0.0.6 The continuous $q$-Hermite polynomials

The continuous $q$-Hermite polynomials are a system of orthogonal polynomials.
Their generating function is

$\sum_{n=0}^{\infty}\frac{H_{n}(\cos(\theta)|q)}{(q;q)_{n}}t^{n}=\frac{1}{(te^{i\theta};q)_{\infty}(te^{-i\theta};q)_{\infty}}\,.$
(A.8)

A useful form of these polynomials is [112]

$H_{n}(\cos\theta|q)\equiv\sum_{k=0}^{n}\binom{n}{k}_{q}e^{i(n-2k)\theta}=\sum_{k=0}^{n}\frac{(q;q)_{n}}{(q;q)_{n-k}(q;q)_{k}}e^{i(n-2k)\theta}\,.$
(A.9)

The $q$-Hermite polynomials satisfy the recursion relation

$2xH_{n}(x|q)=H_{n+1}(x|q)+(1-q^{n})H_{n-1}(x|q)\,,\qquad
H_{-1}(x|q)=0\,,\quad H_{0}(x|q)=1\,.$ (A.10)

They are orthogonal polynomials in $\theta$,

$\int\frac{d\theta}{2\pi}(q,e^{\pm
2i\theta};q)_{\infty}H_{m}(\cos\theta|q)H_{n}(\cos\theta|q)=(q;q)_{n}\delta_{nm}\,,$
(A.11)

and satisfy the relation

$\sum_{n=0}^{\infty}H_{n}(\cos\theta|q)H_{n}(\cos\phi|q)\frac{t^{n}}{(q;q)_{n}}=\frac{(t^{2};q)_{\infty}}{(te^{i(\pm\theta\pm\phi)};q)_{\infty}}\,.$
(A.12)

####  A.0.0.0.7 The $R$-matrix

The $6j$-symbol of $\mathcal{U}_{\sqrt{q}}\left(\mathfrak{su}(1,1)\right)$
takes the explicit form

$R^{(q)}_{\theta_{4}\theta_{2}}\begin{bmatrix}\theta_{3}&l_{2}\\\
\theta_{1}&l_{1}\end{bmatrix}=\frac{\left(\tilde{q}_{1}e^{-i(\theta_{2}+\theta_{3})},\tilde{q}_{1}\tilde{q}_{2}e^{i(\theta_{3}\pm\theta_{1})},\tilde{q}_{1}\tilde{q}_{2}e^{i(\theta_{2}\pm\theta_{4})};q\right)_{\infty}}{\left(\tilde{q}_{1}\tilde{q}_{2}^{2}e^{i(\theta_{2}+\theta_{3})};q\right)_{\infty}}\\\
\times\frac{(\tilde{q}_{2}^{2};q)_{\infty}}{\sqrt{\left(\tilde{q}_{1}e^{i(\pm\theta_{2}\pm\theta_{3})},\tilde{q}_{1}e^{i(\pm\theta_{1}\pm\theta_{4})},\tilde{q}_{2}e^{i(\pm\theta_{1}\pm\theta_{2})},\tilde{q}_{2}e^{i(\pm\theta_{3}\pm\theta_{4})};q\right)_{\infty}}}\\\
\times{}_{8}W_{7}\left(\frac{\tilde{q}_{1}\tilde{q}_{2}^{2}e^{i(\theta_{2}+\theta_{3})}}{q};\tilde{q}_{1}e^{i(\theta_{2}+\theta_{3})},\tilde{q}_{2}e^{i(\theta_{2}\pm\theta_{1})},\tilde{q}_{2}e^{i(\theta_{3}\pm\theta_{4})};q,\tilde{q}_{1}e^{-i(\theta_{2}+\theta_{3})}\right)\,,$
(A.13)

where ${}_{8}W_{7}$ is the basic one-variable well-poised hypergeometric
series,

${}_{8}W_{7}(a;b,c,d,e,f;q,z)=\sum_{n=0}^{\infty}\frac{\left(a,\pm
qa^{1/2},b,c,d,e,f;q\right)_{n}}{\left(\pm
a^{1/2},qa/b,qa/c,qa/d,qa/e,qa/f,q;q\right)_{n}}z^{n}\,.$ (A.14)

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