#  Periodicity of Grover walks on bipartite regular graphs with at most five
distinct eigenvalues

Sho Kubota  Department of Applied Mathematics, Faculty of Engineering,
Yokohama National University, Hodogaya, Yokohama 240-8501, Japan. kubota-sho-
<EMAIL_ADDRESS>
###### Abstract

We determine connected bipartite regular graphs with four distinct adjacency
eigenvalues that induce periodic Grover walks, and show that it is only
$C_{6}$. We also show that there are only three kinds of the second largest
eigenvalues of bipartite regular periodic graphs with five distinct
eigenvalues. Using walk-regularity, we enumerate feasible spectra for such
graphs.
Keywords: Grover walk, quantum walk, periodicity, walk-regular
MSC 2020 subject classifications: 05C50; 81Q99

##  1 Introduction

Quantum walks are quantum analogues of classical random walks [1, 2, 12]. A
great deal of research on quantum walks has been conducted in the last 20
years. There is also a wide range of related fields. In quantum information,
quantum walk models can be seen as generalizations of Grover’s search
algorithm [11, 30]. More recently, quantum cryptography protocols based on
quantum walks have been proposed [29, 35].

The subject of this paper is periodicity of quantum walks. Periodicity has
been studied as one of main problems of quantum walks, and there are several
previous studies. Table 1 summarizes previous studies on periodicity of Grover
walks on undirected graphs. Other models have been studied in [26, 31, 32] for
example. Periodicity is a special case of state transfer problems. The authors
in [23] have applied periodicity to the study of perfect state transfer. In
context of quantum cryptography, periodicity of quantum walks can be a focus
of attention [29].

Graphs | Ref.
---|---
Paths and Cycles | Trivial (or [27])
Complete graphs, complete bipartite graphs, SRGs | [14]
Generalized Bethe trees | [25]
Hamming graphs, Johnson graphs | [36]
Cycles (3-state) | [18]
Complete graphs with self loops | [16]
Table 1: Previous works on periodicity of Grover walks on undirected graphs

In this paper, we consider bipartite regular graphs with at most five distinct
adjacency eigenvalues that induce periodic Grover walks. As we will see in the
beginning of Section 3, the graphs with at most three distinct adjacency
eigenvalues have been substantially studied before. We will therefore study
periodicity of the graphs with four or five distinct adjacency eigenvalues.
There are two main theorems. See later sections for terminologies and
definitions. The first main result states that if a bipartite regular graph
with four or five distinct adjacency eigenvalues is periodic, then the second
largest eigenvalue can only take three different values. The second main
result is that the only bipartite regular graph with four distinct adjacency
eigenvalues to induce periodic Grover walk is $C_{6}$:

######  Theorem 1.1.

Let $\Gamma$ be a bipartite $k$-regular graph with the $A$-spectrum $\\{[\pm
k]^{1},[\pm\theta]^{a},[0]^{b}\\}$, where $a\geq 1$ and $b\geq 0$. Then
$\Gamma$ is periodic if and only if $k$ is even and
$\theta\in\\{\frac{k}{2},\frac{\sqrt{2}}{2}k,\frac{\sqrt{3}}{2}k\\}$.

######  Theorem 1.2.

Let $\Gamma$ be a bipartite $k$-regular graph with the $A$-spectrum $\\{[\pm
k]^{1},[\pm\theta]^{\frac{n}{2}-1}\\}$, where $n$ is the number of vertices of
$\Gamma$. Then $\Gamma$ is periodic if and only if $\Gamma$ is isomorphic to
the cycle graph $C_{6}$.

This paper is organized as follows. Section 2 is mainly preparation. Terms and
facts related to spectral graph theory and Grover walks are introduced. In
Section 3, we derive the first main theorem. Using rings of integers of
quadratic fields, we derive conditions on eigenvalues that periodic graphs
have. In Section 4, we derive the second main theorem. In Section 5, we study
bipartite regular graphs with five distinct adjacency eigenvalues. Focusing on
walk-regularity, we enumerate feasible spectra that periodic graphs have. In
Section 6, we summarize the results and discuss open problems.

##  2 Preliminaries

See [10] for basic terminologies related to graphs. Throughout this paper, we
assume that all graphs are finite, simple, and connected whether we mention or
not. Let $M$ be a square matrix. We call the multiset of eigenvalues of $M$
the spectrum or the $M$-spectrum, and denote it by $\operatorname{Spec}(M)$.
For example, when the $M$-spectrum is $\\{4,0,0,0,-2,-2\\}$, we describe the
multiplicities by superscript, as in $\\{[4]^{1},[0]^{3},[-2]^{2}\\}$. The
matrices $I_{n}$ and $J_{n}$ denote the identity matrix and the all-ones
matrix of size $n$, respectively. The subscripts can be omitted if the sizes
of these matrices are clear in context.

###  2.1 Graphs and their spectra

The adjacency matrix $A=A(\Gamma)\in\mathbb{C}^{V\times V}$ of a graph
$\Gamma=(V,E)$ is defined by

$A_{x,y}=\begin{cases}1\qquad&\text{if $\\{x,y\\}\in E$,}\\\
0\qquad&\text{otherwise.}\end{cases}$

We call eigenvalues of $A(\Gamma)$ adjacency eigenvalues.

######  Proposition 2.1 (Proposition 3.3.1 in [4]).

Let $\Gamma$ be a $k$-regular graph with the adjacency eigenvalues
$k=\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{n}$, where $n$ is the
number of vertices. Then we have $\sum_{i=1}^{n}\lambda_{i}^{2}=nk$.

A graph $\Gamma=(V,E)$ is said to be bipartite if its vertex set can be
partitioned into two parts $V_{1}$ and $V_{2}$ such that for any edge one end
is in $V_{1}$ and the other end is in $V_{2}$. The two parts $V_{1},V_{2}$ are
called partite sets of $V$. It is well-known that the $A$-spectrum of a
bipartite graph is symmetric about the origin and vice versa. See Theorem
8.8.2 in [10] for details. In this paper, we are mainly concerned with
(connected) bipartite regular graphs with four or five distinct adjacency
eigenvalues. The $A$-spectrum of such a graph is usually denoted by
$\\{[k]^{1},[\theta]^{a},[0]^{b},[-\theta]^{a},[-k]^{1}\\}$, but we will
simply write it as $\\{[\pm k]^{1},[\pm\theta]^{a},[0]^{b}\\}$. The following
is a theorem due to Hoffman, often used in studies of graphs with few distinct
adjacency eigenvalues.

######  Proposition 2.2 (Theorem 1 and its proof in [15]).

Let $\Gamma$ be a connected $k$-regular graph with $n$ vertices, and let the
distinct adjacency eigenvalues be $k>\lambda_{2}>\dots>\lambda_{s}$. Then we
have

$q(A(\Gamma))=\frac{q(k)}{n}J_{n},$

where $q(x)=\prod_{i=2}^{s}(x-\lambda_{i})$.

###  2.2 Grover walks

Let $\Gamma$ be a graph. Define
$\mathcal{A}=\mathcal{A}(\Gamma)=\\{(x,y),(y,x)\mid\\{x,y\\}\in E(\Gamma)\\}$.
The origin $x$ and terminus $y$ of $a=(x,y)\in\mathcal{A}$ are denoted by
$o(a),t(a)$, respectively. We write the inverse arc of $a$ as $a^{-1}$.

We introduce several matrices on Grover walks for a graph $\Gamma$. The
boundary matrix $d=d(\Gamma)\in\mathbb{C}^{V\times\mathcal{A}}$ is defined by
$d_{x,a}=\frac{1}{\sqrt{\deg x}}\delta_{x,t(a)}$, where $\delta_{a,b}$ is the
Kronecker delta. The shift matrix
$S=S(\Gamma)\in\mathbb{C}^{\mathcal{A}\times\mathcal{A}}$ is defined by
$S_{a,b}=\delta_{a,b^{-1}}$. Define the time evolution matrix
$U=U(\Gamma)\in\mathbb{C}^{\mathcal{A}\times\mathcal{A}}$ by $U=S(2d^{*}d-I)$.
Quantum walks defined by $U$ is called Grover walks. The discriminant
$T=T(\Gamma)\in\mathbb{C}^{V\times V}$ is defined by $T=dSd^{*}$. Henceforth
we will consider not only the spectrum of the adjacency matrix, but also the
ones of the discriminant and the time evolution matrix. Define
$\operatorname{Spec}_{A}(\Gamma)=\operatorname{Spec}(A(\Gamma))$ for a graph
$\Gamma$. $\operatorname{Spec}_{T}(\Gamma)$ and
$\operatorname{Spec}_{U}(\Gamma)$ are defined in the same way.

######  Lemma 2.3.

Let $\Gamma$ be a $k$-regular graph, and let $A$ and $T$ be the adjacency
matrix and the discriminant of $\Gamma$, respectively. Then we have
$T=\frac{1}{k}A$. Therefore, the absolute values of eigenvalues of $T$ does
not exceed $1$.

We omit a proof. See Section 3 in [24] for more general claim and its proof.
Relationship between $U$-spectra and $T$-spectra has been studied not only in
the Grover walks but also in more general models [13, 20, 22]. We cite the
result in [13], but the statement is slightly modified to fit our setting.

######  Theorem 2.4 ([13]).

Let $\Gamma=(V,E)$ be a connected graph. Then we have

$\operatorname{Spec}_{U}(\Gamma)=\\{e^{\pm
i\arccos\lambda}\mid\lambda\in\operatorname{Spec}_{T}(\Gamma)\\}\cup\\{1\\}^{M_{1}}\cup\\{-1\\}^{M_{-1}},$

where $M_{1}=|E|-|V|+1$ and $M_{-1}=|E|-|V|+\dim\operatorname{Ker}(T+I)$.

By Lemma 2.3 and Theorem 2.4, the $U$-spectrum is obtained from the
$A$-spectrum via the discriminant when a graph is regular.

Let $U$ be the time evolution matrix of a graph $\Gamma$. We say that $\Gamma$
is periodic if there exists a positive integer $\tau$ such that $U^{\tau}=I$.
When a graph $\Gamma$ is periodic, the positive integer
$\min\\{\tau\in\mathbb{N}\mid U^{\tau}=I\\}$ is called the period. As we see
immediately, periodicity is determined by $U$-spectrum.

######  Lemma 2.5 (Lemma 5.3 in [27]).

Let $U$ be the time evolution matrix of a graph $\Gamma$. Then, we have

$\\{\tau\in\mathbb{N}\mid
U^{\tau}=I\\}=\\{\tau\in\mathbb{N}\mid\lambda^{\tau}=1\text{ \emph{for any}
$\lambda\in\operatorname{Spec}_{U}(\Gamma)$}\\}.$

In particular, $\Gamma$ is periodic if and only if there exists a positive
integer $\tau$ such that $\lambda^{\tau}=1$ for any eigenvalue $\lambda$ of
$U$.

##  3 Bipartite regular graphs with at most five distinct adjacency
eigenvalues

In this section, we derive a general fact on bipartite regular periodic graphs
with four or five distinct adjacency eigenvalues. Note that the bipartite
regular graph with two distinct adjacency eigenvalues is the complete graph
$K_{2}$, which is known to be periodic [14]. Bipartite regular graphs with
three distinct adjacency eigenvalues have the $A$-spectra of the form
$\\{[k]^{1},[0]^{n-2},[-k]^{1}\\}$. It is well-known that connected regular
graphs with three distinct adjacency eigenvalues are strongly regular.
Recovering the parameters from the eigenvalues, we see that their complements
are disconnected. Disconnected strongly regular graphs are disjoint unions of
the complete graphs with the same size (See Lemma 10.1.1 in [10]). This
implies that the graphs are the complete bipartite graphs $K_{k,k}$, which are
periodic shown in [14]. In the end, we are concerned with bipartite regular
graphs with four or five distinct adjacency eigenvalues.

Let $\Gamma$ be a bipartite $k$-regular graph with $n$ vertices and the
$A$-spectrum $\\{[\pm k]^{1},[\pm\theta]^{a},[0]^{b}\\}$, where $a\geq 1$ and
$b\geq 0$. Proposition 2.1 derives $2k^{2}+2a\theta^{2}=nk$. We may assume
that $\theta>0$ without loss of generality, and we have

$\theta=\sqrt{\frac{nk-2k^{2}}{2a}}.$ (3.1)

On the other hand, let the partite set of the vertex set be $V_{1}$ and
$V_{2}$, then we obtain

$|V_{1}|=|V_{2}|.$ (3.2)

Indeed, since $\Gamma$ is bipartite, we can display as

$A(\Gamma)=\begin{bmatrix}O&N\\\ N^{\top}&O\end{bmatrix}$

for some $p\times q$ matrix $N$. The graph $\Gamma$ is $k$-regular, so
$N{\bm{1}}_{q}=k{\bm{1}}_{p}$ and $N^{\top}{\bm{1}}_{p}=k{\bm{1}}_{q}$, where
${\bm{1}}_{r}$ denotes the all-ones vector of size $r$. We have
$kp={\bm{1}}_{p}^{\top}(N{\bm{1}}_{q})=({\bm{1}}_{p}^{\top}N){\bm{1}}_{q}=kq$,
i.e., $p=q$. This implies $|V_{1}|=|V_{2}|$.

A complex number $\alpha$ is said to be an algebraic integer if there exists a
monic polynomial $p(x)\in\mathbb{Z}[x]$ such that $p(\alpha)=0$. Let $\Omega$
denote the set of algebraic integers. Note that since the characteristic
polynomial of the adjacency matrix is a monic polynomial with integer
coefficients, the adjacency eigenvalues of a graph $\Gamma$ are algebraic
integers, i.e.,

$\operatorname{Spec}_{A}(\Gamma)\subset\Omega.$ (3.3)

It is well-known that $\Omega$ is a subring of $\mathbb{C}$, and
$\Omega\cap\mathbb{Q}=\mathbb{Z}$. See [17, 28] for algebraic integers. In
addition, we will use integral bases of quadratic fields. A positive integer
$m>1$ is said to be square-free if it is not divisible by $p^{2}$ for any
prime number $p$.

######  Proposition 3.1 (Proposition 2.34 in [17]).

Let $m>1$ be a square-free integer. Then

$\Omega\cap\mathbb{Q}(\sqrt{m})=\begin{cases}\\{p+q\sqrt{m}\mid
p,q\in\mathbb{Z}\\}\quad&\text{if $m\equiv 2,3\pmod{4}$},\\\
\\{p+\frac{1+\sqrt{m}}{2}q\mid p,q\in\mathbb{Z}\\}\quad&\text{if $m\equiv
1\pmod{4}$}.\end{cases}$

Note that $m$ is not square-free when $m\equiv 0\pmod{4}$.

######  Lemma 3.2.

Let $\Gamma$ be a periodic graph. If
$\lambda\in\operatorname{Spec}_{T}(\Gamma)$, then $2\lambda\in\Omega$.

###### Proof.

By Lemma 2.5, there exists a positive integer $\tau$ such that
$\Lambda^{\tau}=1$ for any $\Lambda\in\operatorname{Spec}_{U}(\Gamma)$. The
eigenvalue $\Lambda$ is a root of the monic polynomial
$x^{\tau}-1\in\mathbb{Z}[x]$, so it is an algebraic integer. Let
$\lambda\in\operatorname{Spec}_{T}(\Gamma)$. Theorem 2.4 derives $e^{\pm
i\arccos\lambda}\in\operatorname{Spec}_{U}(\Gamma)$. Since $\Omega$ is a ring,
we have $2\lambda=e^{i\arccos\lambda}+e^{-i\arccos\lambda}\in\Omega$. ∎

######  Theorem 3.3.

Let $\Gamma$ be a bipartite $k$-regular graph with the $A$-spectrum $\\{[\pm
k]^{1},[\pm\theta]^{a},[0]^{b}\\}$, where $a\geq 1$ and $b\geq 0$. Then
$\Gamma$ is periodic if and only if $k$ is even and
$\theta\in\\{\frac{k}{2},\frac{\sqrt{2}}{2}k,\frac{\sqrt{3}}{2}k\\}$.

###### Proof.

First, we show the sufficient condition for $\theta=\frac{k}{2}$. The other
cases are shown in the same way. By Lemma 2.3, we have
$\operatorname{Spec}_{T}(\Gamma)=\\{[\pm
1]^{1},[\pm\frac{1}{2}]^{a},[0]^{b}\\}$. Theorem 2.4 implies
$\operatorname{Spec}_{U}(\Gamma)\subset\\{\pm
1,e^{\pm\frac{\pi}{3}i},e^{\pm\frac{\pi}{2}i}\\}$, where the multiplicities
are ignored. Thus, there exists a positive integer $\tau$ such that
$\lambda^{\tau}=1$ for any $\lambda\in\operatorname{Spec}_{U}(\Gamma)$, and
hence $\Gamma$ is periodic by Lemma 2.5.

We show the necessary condition. We may assume that $\theta>0$ without loss of
generality. We have $\frac{\theta}{k}\in\operatorname{Spec}_{T}(\Gamma)$ since
$\Gamma$ is $k$-regular. Lemma 3.2 derives $\frac{2\theta}{k}\in\Omega$.

Consider the case $\theta\in\mathbb{Q}$. We have
$\frac{2\theta}{k}\in\Omega\cap\mathbb{Q}=\mathbb{Z}$. Lemma 2.3 implies
$0<\frac{\theta}{k}<1$. Since $\frac{2\theta}{k}$ is an integer, we have
$\frac{2\theta}{k}=1$, i.e., $\theta=\frac{k}{2}$. From (3.3), we have
$\theta\in\Omega\cap\mathbb{Q}=\mathbb{Z}$, and hence $k$ is even.

Next, we consider the case $\theta\not\in\mathbb{Q}$. By (3.1), there exist a
square-free integer $m>1$ and $r\in\mathbb{Q}$ such that $\theta=r\sqrt{m}$.
Thus, we have

$\frac{2\theta}{k}=\frac{2r}{k}\sqrt{m}\in\Omega\cap\mathbb{Q}(\sqrt{m}).$
(3.4)

We claim that $m\not\equiv 1\pmod{4}$. Suppose $m\equiv 1\pmod{4}$. By
Proposition 3.1, there exists $p,q\in\mathbb{Z}$ such that

$\frac{2r}{k}\sqrt{m}=p+\frac{1+\sqrt{m}}{2}q.$

Since $1,\sqrt{m}$ are linearly independent over $\mathbb{Q}$, we have

$p+\frac{q}{2}=0$ (3.5)

and

$\frac{2r}{k}=\frac{q}{2}.$ (3.6)

Since $0<\frac{2\theta}{k}<2$, Equalities (3.4) and (3.6) imply
$0<q\sqrt{m}<4$. Since $m\geq 5$, we have $0<q<\frac{4}{\sqrt{m}}<2$, and
hence $q=1$. Equality (3.5) shows that $p=-\frac{1}{2}$, which contradicts to
$p\in\mathbb{Z}$. Now, we have $m\equiv 2,3\pmod{4}$. By Proposition 3.1,
there exists $p,q\in\mathbb{Z}$ such that $\frac{2r}{k}\sqrt{m}=p+q\sqrt{m}$.
Since $1,\sqrt{m}$ are linearly independent over $\mathbb{Q}$, we have $p=0$
and

$q=\frac{2r}{k}.$ (3.7)

Since $0<\frac{2\theta}{k}<2$, Equalities (3.4) and (3.7) imply
$0<q\sqrt{m}<2$. Since $m\geq 2$, we have $0<q<\frac{2}{\sqrt{m}}<2$. This
implies $q=1$, and hence $(q,m)=(1,2),(1,3)$. Therefore, we have
$\frac{2\theta}{k}=\frac{2r}{k}\sqrt{m}=q\sqrt{m}\in\\{\sqrt{2},\sqrt{3}\\}$,
i.e., $\theta\in\\{\frac{\sqrt{2}}{2}k,\frac{\sqrt{3}}{2}k\\}$. The eigenvalue
$\theta$ is also in $\Omega\cap\mathbb{Q}(\sqrt{m})$, so $k$ must be even from
Proposition 3.1. ∎

##  4 Four distinct adjacency eigenvalues

In this section, we consider a bipartite regular graph $\Gamma$ with four
distinct adjacency eigenvalues. As mentioned in Proposition 15.1.3 in [4], it
is known that such a graph is the incidence graph of a symmetric 2-design.
Therefore there are a very large number of connected bipartite regular graphs
with four distinct adjacency eigenvalues, of which we will show that only
$C_{6}$ is periodic. Let the $A$-spectrum of $\Gamma$ be $\\{[\pm
k]^{1},[\pm\theta]^{a}\\}$, where $a\geq 1$. Let $n$ be the number of vertices
of $\Gamma$. Then we have $a=\frac{n}{2}-1$. From the previous section, if
$\Gamma$ is periodic, then
$\theta\in\\{\frac{k}{2},\frac{\sqrt{2}}{2}k,\frac{\sqrt{3}}{2}k\\}$.

######  Theorem 4.1.

Let $\Gamma$ be a bipartite $k$-regular graph with the $A$-spectrum $\\{[\pm
k]^{1},[\pm\theta]^{\frac{n}{2}-1}\\}$, where $n$ is the number of vertices of
$\Gamma$. Then $\Gamma$ is periodic if and only if $\Gamma$ is isomorphic to
the cycle graph $C_{6}$.

###### Proof.

It is well-known that the cycle graph $C_{6}$ has the $A$-spectrum $\\{[\pm
2]^{1},[\pm 1]^{2}\\}$, which is periodic. We show that the assumed periodic
graph is determined to $C_{6}$. Since $\Gamma$ is bipartite, its adjacency
matrix is displayed as

$A=A(\Gamma)=\begin{bmatrix}O&N\\\ N^{\top}&O\end{bmatrix}$

for some matrix $N$. This matrix $N$ is square matrix of size $\frac{n}{2}$
from (3.2). We have

$\displaystyle\\{[k^{2}]^{2},[\theta^{2}]^{n-2}\\}$
$\displaystyle=\operatorname{Spec}(A^{2})$
$\displaystyle=\operatorname{Spec}(NN^{\top})\cup\operatorname{Spec}(N^{\top}N)$
$\displaystyle=\operatorname{Spec}(NN^{\top})\cup\operatorname{Spec}(NN^{\top}).$

This implies
$\operatorname{Spec}(NN^{\top})=\\{[k^{2}]^{1},[\theta^{2}]^{\frac{n}{2}-1}\\}$,
that is,

$\operatorname{Spec}(NN^{\top}-\theta^{2}I)=\\{[k^{2}-\theta^{2}]^{1},[0]^{\frac{n}{2}-1}\\}.$
(4.1)

We take an eigenvector $\sqrt{\frac{2}{n}}{\bm{1}}_{\frac{n}{2}}$ of norm $1$
associated to the eigenvalue $k^{2}-\theta^{2}$, and eigenvectors
$x_{1},\dots,x_{\frac{n}{2}-1}$ associated to the eigenvalue $0$ such that
$x_{i}^{\top}x_{j}=\delta_{i,j}$. The matrix
$P=[\sqrt{\frac{2}{n}}{\bm{1}}_{\frac{n}{2}},x_{1},\dots,x_{\frac{n}{2}-1}]$
is an orthogonal matrix and diagonalizes $NN^{\top}-\theta^{2}I$. Thus we have

$NN^{\top}-\theta^{2}I=P\operatorname{diag}(k^{2}-\theta^{2},0,\dots,0)P^{\top}=\frac{2(k^{2}-\theta^{2})}{n}J_{\frac{n}{2}},$

that is,

$NN^{\top}=\theta^{2}I+\frac{2(k^{2}-\theta^{2})}{n}J_{\frac{n}{2}}$ (4.2)

Since $\Gamma$ is $k$-regular and $N$ is $\\{0,1\\}$-matrix, we have

$(NN^{\top})_{x,x}=\sum_{z\in V(\Gamma)}N_{x,z}N_{x,z}=\sum_{z\in
V(\Gamma)}N_{x,z}=k$

for any vertex $x\in V(\Gamma)$. The $(x,x)$ entries of both sides of Equality
(4.2) are $k=\theta^{2}+\frac{2(k^{2}-\theta^{2})}{n}$. In particular, we have

$k>\theta^{2}.$ (4.3)

Suppose $\theta\in\\{\frac{\sqrt{2}}{2}k,\frac{\sqrt{3}}{2}k\\}$. Since
$\theta\geq\frac{\sqrt{2}}{2}k$, Inequality (4.3) derives $k<2$. However, $k$
is even by Theorem 3.3. This is impossible. Thus $\theta=\frac{k}{2}$. By
(4.3) again, we have $k<4$, and hence $k=2$ because $k$ is even. The
$A$-spectrum of $\Gamma$ is determined to be $\\{[\pm 2]^{1},[\pm
1]^{\frac{n}{2}-1}\\}$. Equality (3.1) implies $n=6$. Since $\Gamma$ is
connected, the graph is determined to be the cycle graph $C_{6}$. ∎

##  5 Five distinct adjacency eigenvalues

In contrast to bipartite regular graphs with four distinct adjacency
eigenvalues, we can construct a large number of periodic graphs with five
distinct adjacency eigenvalues. On the other hand, of the feasible spectra,
there are many graphs whose existence is unknown. Using tools of spectral
graph theory, we discuss feasible periodic graphs with five distinct adjacency
eigenvalues.

###  5.1 Construction

Let $\Gamma$ be a graph. We write $\Gamma\otimes J_{m}$ as the graph defined
by $A(\Gamma\otimes J_{m})=A(\Gamma)\otimes J_{m}$. If $m=1$, we simply regard
$\Gamma\otimes J_{1}$ as $\Gamma$. If a graph $\Gamma$ is $k$-regular and has
the $A$-spectrum
$\\{[k]^{1},[\lambda_{1}]^{f_{1}},\dots,[\lambda_{s}]^{f_{s}}\\}$, then the
graph $\Gamma\otimes J_{m}$ is $km$-regular, and

$\displaystyle\operatorname{Spec}_{A}(\Gamma\otimes J_{m})$
$\displaystyle=\\{[km]^{1},[\lambda_{1}m]^{f_{1}},\dots,[\lambda_{s}m]^{f_{s}},[0]^{nm-n}\\},$
$\displaystyle\operatorname{Spec}_{T}(\Gamma\otimes J_{m})$
$\displaystyle=\left\\{[1]^{1},\left[\frac{\lambda_{1}}{k}\right]^{f_{1}},\dots,\left[\frac{\lambda_{s}}{k}\right]^{f_{s}},[0]^{nm-n}\right\\},$

where $n$ is the number of vertices of $\Gamma$. In particular, if $A(\Gamma)$
has the eigenvalue 0, then $T(\Gamma)$ and $T(\Gamma\otimes J_{m})$ have the
same spectrum except for the multiplicities. From this observation, it follows
that $C_{6}\otimes J_{m}$ and $C_{8}\otimes J_{m}$ are also periodic graphs.
Note that the $U$-spectrum of $C_{6}\otimes J_{m}$ ignoring its multiplicities
is $\\{\pm 1,\pm i,e^{\pm\frac{\pi}{3}i},e^{\pm\frac{2\pi}{3}i}\\}$ from
Theorem 2.4. Thus, the period is $12$. Similarly, the period of $C_{8}\otimes
J_{m}$ is $8$.

Another construction method is to use graph products. Let $\Gamma$ and
$\Delta$ be graphs with $n$ and $m$ vertices, respectively. The Cartesian
product $\Gamma\mathrel{\square}\Delta$ is the graph defined by
$A(\Gamma\mathrel{\square}\Delta)=A(\Gamma)\otimes I_{m}+I_{n}\otimes
A(\Delta)$. The Kronecker product $\Gamma\otimes\Delta$ is the graph defined
by $A(\Gamma\otimes\Delta)=A(\Gamma)\otimes A(\Delta)$. In particular, the
graph $\Gamma\otimes K_{2}$ is called the bipartite double of $\Gamma$. The
$A$-spectrum of $\Gamma\otimes K_{2}$ is
$\operatorname{Spec}_{A}(\Gamma)\cup-\operatorname{Spec}_{A}(\Gamma)$. These
graph products can also be used to construct periodic graphs. For example, the
line graph of the $3$-dimensional hypercube $Q_{3}$, i.e., $L(Q_{3})$ has the
$A$-spectrum $\\{[4]^{1},[2]^{3},[0]^{3},[-2]^{5}\\}$. Thus, its bipartite
double $L(Q_{3})\otimes K_{2}$ has the $A$-spectrum $\\{[\pm 4]^{1},[\pm
2]^{8},[0]^{6}\\}$, which is a connected bipartite regular periodic graph with
five distinct adjacency eigenvalues. For more information on graph products,
see [4, 21].

###  5.2 Feasible spectra

From Theorem 3.3, if a bipartite $k$-regular graph $\Gamma$ with five distinct
adjacency eigenvalues is periodic, the second largest eigenvalue can only take
one of the three values, and $k$ must be even. However, there are still
infinite possibilities for the number of vertices and multiplicities of
eigenvalues. We need to narrow down candidates for $A$-spectra of graphs.

######  Lemma 5.1.

Let $\Gamma$ be a bipartite $k$-regular graph with the $A$-spectrum $\\{[\pm
k]^{1},[\pm\theta]^{a},[0]^{b}\\}$, where $a,b\geq 1$. If we display the
adjacency matrix $A$ of $\Gamma$ as

$A=\begin{bmatrix}O&N\\\ N^{\top}&O\end{bmatrix},$

then we have
$NN^{\top}N=\theta^{2}N+\frac{2k}{n}(k^{2}-\theta^{2})J_{\frac{n}{2}}$, where
$n$ is the number of vertices.

###### Proof.

The two vectors

$\alpha=\frac{1}{\sqrt{n}}\begin{bmatrix}{\bm{1}}_{\frac{n}{2}}\\\
{\bm{1}}_{\frac{n}{2}}\end{bmatrix},\qquad\beta=\frac{1}{\sqrt{n}}\begin{bmatrix}{\bm{1}}_{\frac{n}{2}}\\\
-{\bm{1}}_{\frac{n}{2}}\end{bmatrix}$

are eigenvectors of $A$ associated to $k,-k$ with norm $1$, respectively. Let
$x_{1},\dots,x_{n-2}$ be eigenvectors of $A$ associated to
$\lambda\not\in\\{k,-k\\}$ such that $x_{i}^{\top}x_{j}=\delta_{i,j}$. Define
$M=(A^{2}-\theta^{2}I_{n})A$. Then $x_{1},\dots,x_{n-2}$ are eigenvectors of
$M$ associated to $0$. Consider $Q=[\alpha,\beta,x_{1},\dots,x_{n-2}]$. The
matrix $Q$ is an orthogonal matrix and diagonalizes $M$, so we have

$\displaystyle M$
$\displaystyle=Q\operatorname{diag}\left((k^{2}-\theta^{2})k,-(k^{2}-\theta^{2})k,0\dots,0\right)Q^{\top}$
$\displaystyle=(k^{2}-\theta^{2})k(\alpha\alpha^{\top}-\beta\beta^{\top})$
$\displaystyle=\frac{2k(k^{2}-\theta^{2})}{n}\begin{bmatrix}O&J_{\frac{n}{2}}\\\
J_{\frac{n}{2}}&O\end{bmatrix}.$

On the other hand,

$\displaystyle M$ $\displaystyle=\left(A^{2}-\theta^{2}I_{n}\right)A$
$\displaystyle=\left(\begin{bmatrix}O&N\\\
N^{\top}&O\end{bmatrix}^{2}-\theta^{2}I_{n}\right)\begin{bmatrix}O&N\\\
N^{\top}&O\end{bmatrix}$
$\displaystyle=\begin{bmatrix}O&NN^{\top}N-\theta^{2}N\\\
N^{\top}NN^{\top}-\theta^{2}N^{\top}&O\end{bmatrix}.$

Comparing (1,2)-block, we obtain the statement. ∎

######  Lemma 5.2.

With the above notation, we have $n\leq 2k(k^{2}-\theta^{2})$.

###### Proof.

Pick $x\in V(\Gamma)$. Let $\Gamma_{j}(x)$ be the set of vertices at distance
$j$ from $x$. Display as $\Gamma_{2}(x)=\\{y_{1},\dots,y_{t}\\}$. Suppose that
$\Gamma_{1}(x)=\Gamma_{1}(y_{i})$ for any $y_{i}\in\Gamma_{2}(x)$. Since
$\Gamma$ is connected, $\Gamma$ is isomorphic to the complete bipartite graph
$K_{k,k}$. However, this contradicts the assumption of the $A$-spectrum of
$\Gamma$. Thus, there exists $y_{i}\in\Gamma_{2}(x)$ such that
$\Gamma_{1}(y_{i})\neq\Gamma_{1}(x)$. Since $\Gamma$ is $k$-regular, neither
$\Gamma_{1}(y_{i})$ nor $\Gamma_{1}(x)$ is included in the other. Thus, we
have $\Gamma_{1}(y_{i})\setminus\Gamma_{1}(x)\neq\emptyset$. Pick
$z\in\Gamma_{1}(y_{i})\setminus\Gamma_{1}(x)$. The pair of vertices $x,z$
satisfies $N_{x,z}=0$ and $(NN^{\top}N)_{x,z}\geq 1$. By Lemma 5.1,

$1\leq(NN^{\top}N)_{x,z}=\left(\theta^{2}N+\frac{2k}{n}(k^{2}-\theta^{2})J_{\frac{n}{2}}\right)_{x,z}=\frac{2k}{n}(k^{2}-\theta^{2}).$

We have the statement. ∎

A graph $\Gamma$ with the adjacency matrix $A$ is said to be walk-regular if
$(A^{r})_{x,x}$ is independent of the choice of $x$ for each positive integer
$r$. Walk-regular graphs are regular since $\deg x=(A^{2})_{x,x}$ is a
constant. See [9] for more information on walk-regular graphs. Walk-regularity
provides conditions for the existence of graphs. Let the constant
$(A^{r})_{x,x}$ be $c_{r}$ for a positive integer $r$, and let the adjacency
eigenvalues be $k=\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{n}$. We have

$\sum_{i=1}^{n}\lambda_{i}^{r}=\operatorname{tr}(A^{r})=\sum_{x\in
V(\Gamma)}(A^{r})_{x,x}=nc_{r},$

where $n$ is the number of vertices. The constant $c_{r}$ is the number of
closed walks, so

$c_{r}=\frac{1}{n}\sum_{i=1}^{n}\lambda_{i}^{r}$

is a non-negative integer. This condition restricts feasible spectra of walk-
regular graphs. Van Dam pointed out that regular graphs with four adjacency
eigenvalues are walk-regular [5]. Koledin and Stanić showed that regular
bipartite graphs with three distinct non-negative adjacency eigenvalues are
walk-regular [19]. Although a proof is similar, the graphs we consider are
also walk-regular.

######  Lemma 5.3.

Let $\Gamma$ be a bipartite $k$-regular graph with the $A$-spectrum $\\{[\pm
k]^{1},[\pm\theta]^{a},[0]^{b}\\}$, where $a,b\geq 1$. Then $\Gamma$ is walk-
regular.

###### Proof.

Let $n$ be the number of vertices of $\Gamma$, and let $A$ be the adjacency
matrix. By Proposition 2.2, we have $q(A)=\frac{q(k)}{n}J_{n}$, where
$q(x)=x(x+k)(x^{2}-\theta^{2})$. In particular, $A^{4}$ can be expressed as a
linear combination of $A^{3},A^{2},A,I_{n}$, and $J_{n}$ with rational
coefficients. Thus for any positive integer $r$, there exist
$\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5}\in\mathbb{Q}$ such
that
$A^{r}=\alpha_{1}A^{3}+\alpha_{2}A^{2}+\alpha_{3}A+\alpha_{4}I_{n}+\alpha_{5}J_{n}$.
On the other hand, $(A^{3})_{x,x}=A_{x,x}=0$ for any $x\in V(\Gamma)$ since
$\Gamma$ is bipartite. We have
$(A^{r})_{x,x}=k\alpha_{2}+\alpha_{4}+\alpha_{5}$, and hence $\Gamma$ is walk-
regular. ∎

######  Proposition 5.4.

Let $\Gamma$ be a bipartite $k$-regular graph with $n$ vertices. Suppose that
the $A$-spectrum is $\operatorname{Spec}_{A}(\Gamma)=\\{[\pm
k]^{1},[\pm\theta]^{a},[0]^{b}\\}$, where $a,b\geq 1$. Then we have the
following.

  1. (i)

$a=\frac{nk-2k^{2}}{2\theta^{2}}$ and $b=n-2-\frac{nk-2k^{2}}{\theta^{2}}$. In
particular, $\frac{nk-2k^{2}}{2\theta^{2}}$ and
$n-2-\frac{nk-2k^{2}}{\theta^{2}}$ are positive integers;

  2. (ii)

$\frac{2(k^{2}+\theta^{2})}{k}\leq n\leq 2k(k^{2}-\theta^{2})$; and

  3. (iii)

For any positive integer $r$, we have $\frac{1}{n}(2\cdot
k^{2r}+(nk-2k^{2})\theta^{2r-2})\in\mathbb{Z}$.

###### Proof.

(i) Since the sum of the multiplicities is the number of vertices, we have

$2a+b+2=n.$ (5.1)

Proposition 2.1 implies

$2k^{2}+2a\theta^{2}=nk,$ (5.2)

which leads to $a=\frac{nk-2k^{2}}{2\theta^{2}}$. Equalities (5.1) and (5.2)
imply $b=n-2-\frac{nk-2k^{2}}{\theta^{2}}$. Since the multiplicities of
eigenvalues are positive integers, $\frac{nk-2k^{2}}{2\theta^{2}}$ and
$n-2-\frac{nk-2k^{2}}{\theta^{2}}$ are also positive integers.

(ii) Since $a\geq 1$, we have $\frac{nk-2k^{2}}{2\theta^{2}}\geq 1$, i.e.,
$n\geq\frac{2(k^{2}+\theta^{2})}{k}$. On the other hand, Lemma 5.2 derives
$n\leq 2k(k^{2}-\theta^{2})$. Thus, we have $\frac{2(k^{2}+\theta^{2})}{k}\leq
n\leq 2k(k^{2}-\theta^{2})$.

(iii) By Lemma 5.3, the graph $\Gamma$ is walk-regular. Thus, the constant
$c_{2r}$ is a non-negative integer for any positive integer $r$. We have
$\frac{1}{n}(2\cdot k^{2r}+(nk-2k^{2})\theta^{2r-2})\in\mathbb{Z}$. ∎

The above proposition narrows down the candidates of graphs. Indeed, the
condition (ii) makes possibility of $n$ finite when $k$ is fixed. The
conditions (i) and (iii) further restrict possibilities for $n$ and $k$. For
convenience, we call graphs to pass Proposition 5.4 feasible periodic graphs.
Table 2, Table 3, and Table 4, which are shown after references for reasons of
space, list feasible periodic graphs. In these tables, there are many graphs
whose existence is unknown. The column “Existence” gives one example that
realizes the spectrum if such graphs exist. See [3] for the symbols of the
graphs. In addition, the symbol “$-$” in the tables denotes that the spectrum
passes Proposition 5.4 but the non-existence of a graph is shown by other
reasons. The reasons are briefly described in the rightmost column of the
tables. For example, a graph with $A$-spectrum of the form $\\{[\pm
4]^{1},[\pm 2]^{a},[0]^{b}\\}$ is a 4-regular integral graph. Thus, the
classification by Stevanović [33] apply, and hence there is no such graph with
more than $32$ vertices.

###  5.3 Quadrangles

We next focus on the sum of the fourth power of the eigenvalues. This contains
the information of quadrangles in graphs. Such geometric information gives us
a slightly stronger condition for the existence of graphs. The following is
substantially pointed out also in [5, 34], but we give a proof for wider
readers.

######  Lemma 5.5.

Let $\Gamma$ be a walk-regular graph with the adjacency eigenvalues
$k=\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{n}$. Denote by $q$ the
number of quadrangles in $\Gamma$ and by $q_{x}$ the number of quadrangles
containing a vertex $x\in V(\Gamma)$. Then we have

  1. (i)

$\sum_{i=1}^{n}\lambda_{i}^{4}=n(2k^{2}-k)+8q$; and

  2. (ii)

The number $q_{x}$ is a constant independent of the choice of a vertex, and we
have $q_{x}=\frac{4q}{n}$.

###### Proof.

Let the adjacency matrix of $\Gamma$ be $A$. First, we observe the $(x,x)$
entry of $A^{4}$ for a vertex $x$. It represents the number of closed walks of
length $4$ from $x$ to $x$. See Figure 1. There are four possibilities for a
closed walk of length $4$. Case (a) shows a closed walk $(x,y_{1},x,y_{1},x)$
where only one vertex $y_{1}$ appears besides $x$. There are $k$ ways to get
$y_{1}$, so there are $k$ walks of this type. Case (b) shows a closed walk
$(x,y_{1},x,y_{2},x)$ where two vertices $y_{1},y_{2}$ are adjacent to $x$.
The total number getting $y_{1},y_{2}$ is $k(k-1)$. Thus, there are $k(k-1)$
walks of this type. Case (c) shows a closed walk $(x,y_{1},y_{2},y_{1},x)$
where $y_{1}$ is adjacent to $x$ but $y_{2}$ is not adjacent to $x$. There are
$k$ ways to take a vertex $y_{1}$, and for each of them there are $k-1$ ways
to take $y_{2}$. Thus, there are $k(k-1)$ walks of this type. Case (d) shows a
closed walk that forms a quadrangle. For each quadrangle, there are two closed
walks $(x,y_{1},y_{2},y_{3},x)$ and $(x,y_{3},y_{2},y_{1},x)$. Thus, we get
$2q_{x}$ walks of this type. By the four cases, we have

$(A^{4})_{x,x}=2k^{2}-k+2q_{x}.$ (5.3)

Let the set of the quadrangles in $\Gamma$ be $\mathcal{Q}$. Count
$|\\{(x,Q)\in V(\Gamma)\times\mathcal{Q}\mid x\in Q\\}|$ in two ways, and we
obtain

$\sum_{x\in V(\Gamma)}q_{x}=4q.$ (5.4)

Indeed,

$|\\{(x,Q)\in V(\Gamma)\times\mathcal{Q}\mid x\in Q\\}|=\sum_{x\in
V(\Gamma)}|\\{Q\in\mathcal{Q}\mid x\in Q\\}|=\sum_{x\in V(\Gamma)}q_{x},$

and we have

$|\\{(x,Q)\in V(\Gamma)\times\mathcal{Q}\mid x\in
Q\\}|=\sum_{Q\in\mathcal{Q}}|\\{x\in V(\Gamma)\mid x\in
Q\\}|=\sum_{Q\in\mathcal{Q}}4=4q.$

Thus,

$\sum_{i=1}^{n}\lambda_{i}^{4}=\sum_{x\in V(\Gamma)}(A^{4})_{x,x}=\sum_{x\in
V(\Gamma)}(2k^{2}-k+2q_{x})=n(2k^{2}-k)+8q.$

Equality (5.3) and walk-regularity derive
$c_{4}=(A^{4})_{x,x}=2k^{2}-k+2q_{x}$, so $q_{x}$ is also a constant. Thus,
Equality (5.4) implies $4q=\sum_{x\in V(\Gamma)}q_{x}=nq_{x}$, that is,
$q_{x}=\frac{4q}{n}$. ∎

Case (a)$x$$y_{1}$

Case (b)$x$$y_{1}$$y_{2}$ Case (c)$x$$y_{1}$$y_{2}$ Case
(d)$x$$y_{1}$$y_{2}$$y_{3}$

Figure 1: Closed walks of length $4$

From the above lemma,
$q=\frac{1}{8}(\sum_{i=1}^{n}\lambda_{i}^{4}-n(2k^{2}-k))$ and
$q_{x}=\frac{1}{2n}(\sum_{i=1}^{n}\lambda_{i}^{4}-n(2k^{2}-k))$ must be non-
negative integers. This is another necessary condition different from
Proposition 5.4 for the existence of graphs. Indeed, several feasible periodic
graphs in the tables are eliminated by this observation.

$k$ | $n$ | Spectrum | Existence | Comment
---|---|---|---|---
4 | 12 | $\\{[\pm 4]^{1},[\pm 2]^{2},[0]^{6}\\}$ | $C_{6}\otimes J_{2}$ |
4 | 16 | $\\{[\pm 4]^{1},[\pm 2]^{4},[0]^{6}\\}$ | $H(4,2)$ |
4 | 24 | $\\{[\pm 4]^{1},[\pm 2]^{8},[0]^{6}\\}$ | $L(Q_{3})\otimes K_{2}$ |
4 | 32 | $\\{[\pm 4]^{1},[\pm 2]^{12},[0]^{6}\\}$ | $IG(AG(2,4)\setminus{\rm pc})$ |  $q=0$, [3, 7]
4 | 48 | $\\{[\pm 4]^{1},[\pm 2]^{20},[0]^{6}\\}$ | $-$ | [33]
4 | 64 | $\\{[\pm 4]^{1},[\pm 2]^{28},[0]^{6}\\}$ | $-$ | [33]
4 | 96 | $\\{[\pm 4]^{1},[\pm 2]^{44},[0]^{6}\\}$ | $-$ | [33]
6 | 18 | $\\{[\pm 6]^{1},[\pm 3]^{2},[0]^{12}\\}$ | $C_{6}\otimes J_{3}$ |
6 | 24 | $\\{[\pm 6]^{1},[\pm 3]^{4},[0]^{14}\\}$ | $-$ | $q_{x}\not\in\mathbb{Z}$
6 | 36 | $\\{[\pm 6]^{1},[\pm 3]^{8},[0]^{18}\\}$ | ? |
6 | 54 | $\\{[\pm 6]^{1},[\pm 3]^{14},[0]^{24}\\}$ | $H(3,3)\otimes K_{2}$ |
6 | 72 | $\\{[\pm 6]^{1},[\pm 3]^{20},[0]^{30}\\}$ | $-$ | $q_{x}\not\in\mathbb{Z}$
6 | 108 | $\\{[\pm 6]^{1},[\pm 3]^{32},[0]^{42}\\}$ | ? |
6 | 162 | $\\{[\pm 6]^{1},[\pm 3]^{50},[0]^{60}\\}$ | $IG(pg(5,5,2))$ |  $q=0$, [3, 6]
6 | 216 | $\\{[\pm 6]^{1},[\pm 3]^{68},[0]^{78}\\}$ | $-$ | $q<0$
6 | 324 | $\\{[\pm 6]^{1},[\pm 3]^{104},[0]^{114}\\}$ | $-$ | $q<0$
8 | 24 | $\\{[\pm 8]^{1},[\pm 4]^{2},[0]^{18}\\}$ | $C_{6}\otimes J_{4}$ |
8 | 32 | $\\{[\pm 8]^{1},[\pm 4]^{4},[0]^{22}\\}$ | $H(4,2)\otimes J_{2}$ |
8 | 48 | $\\{[\pm 8]^{1},[\pm 4]^{8},[0]^{30}\\}$ | $L(Q_{3})\otimes K_{2}\otimes J_{2}$ |
8 | 64 | $\\{[\pm 8]^{1},[\pm 4]^{12},[0]^{38}\\}$ | $K_{4,4}\mathrel{\square}K_{4,4}$ |
8 | 96 | $\\{[\pm 8]^{1},[\pm 4]^{20},[0]^{54}\\}$ | ? |
8 | 128 | $\\{[\pm 8]^{1},[\pm 4]^{28},[0]^{70}\\}$ | ? |
8 | 192 | $\\{[\pm 8]^{1},[\pm 4]^{44},[0]^{102}\\}$ | ? |
8 | 256 | $\\{[\pm 8]^{1},[\pm 4]^{60},[0]^{134}\\}$ | ? |
8 | 384 | $\\{[\pm 8]^{1},[\pm 4]^{92},[0]^{198}\\}$ | ? |
8 | 512 | $\\{[\pm 8]^{1},[\pm 4]^{124},[0]^{262}\\}$ | ? |
8 | 768 | $\\{[\pm 8]^{1},[\pm 4]^{188},[0]^{390}\\}$ | ? |
10 | 30 | $\\{[\pm 10]^{1},[\pm 5]^{2},[0]^{24}\\}$ | $C_{6}\otimes J_{5}$ |
10 | 40 | $\\{[\pm 10]^{1},[\pm 5]^{4},[0]^{30}\\}$ | $-$ | $q_{x}\not\in\mathbb{Z}$
10 | 50 | $\\{[\pm 10]^{1},[\pm 5]^{6},[0]^{36}\\}$ | ? |
10 | 60 | $\\{[\pm 10]^{1},[\pm 5]^{8},[0]^{42}\\}$ | ? |
10 | 100 | $\\{[\pm 10]^{1},[\pm 5]^{16},[0]^{66}\\}$ | ? |
10 | 120 | $\\{[\pm 10]^{1},[\pm 5]^{20},[0]^{78}\\}$ | $-$ | $q_{x}\not\in\mathbb{Z}$
10 | 150 | $\\{[\pm 10]^{1},[\pm 5]^{26},[0]^{96}\\}$ | ? |
10 | 200 | $\\{[\pm 10]^{1},[\pm 5]^{36},[0]^{126}\\}$ | $-$ | $q_{x}\not\in\mathbb{Z}$
10 | 250 | $\\{[\pm 10]^{1},[\pm 5]^{46},[0]^{156}\\}$ | ? |
10 | 300 | $\\{[\pm 10]^{1},[\pm 5]^{56},[0]^{186}\\}$ | ? |
10 | 500 | $\\{[\pm 10]^{1},[\pm 5]^{96},[0]^{306}\\}$ | ? |
10 | 600 | $\\{[\pm 10]^{1},[\pm 5]^{116},[0]^{366}\\}$ | $-$ | $q_{x}\not\in\mathbb{Z}$
10 | 750 | $\\{[\pm 10]^{1},[\pm 5]^{146},[0]^{456}\\}$ | ? |
10 | 1000 | $\\{[\pm 10]^{1},[\pm 5]^{196},[0]^{606}\\}$ | $-$ | $q_{x}\not\in\mathbb{Z}$
10 | 1250 | $\\{[\pm 10]^{1},[\pm 5]^{246},[0]^{756}\\}$ | ? |
10 | 1500 | $\\{[\pm 10]^{1},[\pm 5]^{296},[0]^{906}\\}$ | ? |
$\vdots$ | $\vdots$ |  |  |
Table 2: Feasible periodic graphs whose $A$-spectra are the form $\\{[\pm k]^{1},[\pm\frac{k}{2}]^{a},[0]^{b}\\}$. $k$ | $n$ | Spectrum | Existence | Comment
---|---|---|---|---
2 | 8 | $\\{[\pm 2]^{1},[\pm\sqrt{2}]^{2},[0]^{2}\\}$ | $C_{8}$ |
4 | 16 | $\\{[\pm 4]^{1},[\pm 2\sqrt{2}]^{2},[0]^{10}\\}$ | $C_{8}\otimes J_{2}$ |
4 | 32 | $\\{[\pm 4]^{1},[\pm 2\sqrt{2}]^{6},[0]^{18}\\}$ | TD${}_{1}(2,4)\otimes J_{2,1}$ | [8]
4 | 64 | $\\{[\pm 4]^{1},[\pm 2\sqrt{2}]^{14},[0]^{34}\\}$ | ? |
6 | 18 | $\\{[\pm 6]^{1},[\pm 3\sqrt{2}]^{1},[0]^{14}\\}$ | $-$ | $q\not\in\mathbb{Z}$
6 | 24 | $\\{[\pm 6]^{1},[\pm 3\sqrt{2}]^{2},[0]^{18}\\}$ | $C_{8}\otimes J_{3}$ |
6 | 36 | $\\{[\pm 6]^{1},[\pm 3\sqrt{2}]^{4},[0]^{26}\\}$ | ? |
6 | 48 | $\\{[\pm 6]^{1},[\pm 3\sqrt{2}]^{6},[0]^{34}\\}$ | $-$ | $q_{x}\not\in\mathbb{Z}$
6 | 54 | $\\{[\pm 6]^{1},[\pm 3\sqrt{2}]^{7},[0]^{38}\\}$ | $-$ | $q\not\in\mathbb{Z}$
6 | 72 | $\\{[\pm 6]^{1},[\pm 3\sqrt{2}]^{10},[0]^{50}\\}$ | ? |
6 | 108 | $\\{[\pm 6]^{1},[\pm 3\sqrt{2}]^{16},[0]^{74}\\}$ | ? |
6 | 144 | $\\{[\pm 6]^{1},[\pm 3\sqrt{2}]^{22},[0]^{98}\\}$ | $-$ | $q_{x}\not\in\mathbb{Z}$
6 | 162 | $\\{[\pm 6]^{1},[\pm 3\sqrt{2}]^{25},[0]^{110}\\}$ | $-$ | $q\not\in\mathbb{Z}$
6 | 216 | $\\{[\pm 6]^{1},[\pm 3\sqrt{2}]^{34},[0]^{146}\\}$ | ? |
8 | 32 | $\\{[\pm 8]^{1},[\pm 4\sqrt{2}]^{2},[0]^{26}\\}$ | $C_{8}\otimes J_{4}$ |
8 | 64 | $\\{[\pm 8]^{1},[\pm 4\sqrt{2}]^{6},[0]^{50}\\}$ | TD${}_{1}(2,4)\otimes J_{2,1}\otimes J_{2}$ | [8]
8 | 128 | $\\{[\pm 8]^{1},[\pm 4\sqrt{2}]^{14},[0]^{98}\\}$ | ? |
8 | 256 | $\\{[\pm 8]^{1},[\pm 4\sqrt{2}]^{30},[0]^{194}\\}$ | ? |
8 | 512 | $\\{[\pm 8]^{1},[\pm 4\sqrt{2}]^{62},[0]^{386}\\}$ | ? |
10 | 40 | $\\{[\pm 10]^{1},[\pm 5\sqrt{2}]^{2},[0]^{34}\\}$ | $C_{8}\otimes J_{5}$ |
10 | 50 | $\\{[\pm 10]^{1},[\pm 5\sqrt{2}]^{3},[0]^{42}\\}$ | $-$ | $q\not\in\mathbb{Z}$
10 | 80 | $\\{[\pm 10]^{1},[\pm 5\sqrt{2}]^{6},[0]^{66}\\}$ | $-$ | $q_{x}\not\in\mathbb{Z}$
10 | 100 | $\\{[\pm 10]^{1},[\pm 5\sqrt{2}]^{8},[0]^{82}\\}$ | ? |
10 | 200 | $\\{[\pm 10]^{1},[\pm 5\sqrt{2}]^{18},[0]^{162}\\}$ | ? |
10 | 250 | $\\{[\pm 10]^{1},[\pm 5\sqrt{2}]^{23},[0]^{202}\\}$ | $-$ | $q\not\in\mathbb{Z}$
10 | 400 | $\\{[\pm 10]^{1},[\pm 5\sqrt{2}]^{38},[0]^{322}\\}$ | $-$ | $q_{x}\not\in\mathbb{Z}$
10 | 500 | $\\{[\pm 10]^{1},[\pm 5\sqrt{2}]^{48},[0]^{402}\\}$ | ? |
10 | 1000 | $\\{[\pm 10]^{1},[\pm 5\sqrt{2}]^{98},[0]^{802}\\}$ | ? |
$\vdots$ | $\vdots$ |  |  |
Table 3: Feasible periodic graphs whose $A$-spectra are the form $\\{[\pm k]^{1},[\pm\frac{\sqrt{2}}{2}k]^{a},[0]^{b}\\}$. $k$ | $n$ | Spectrum | Existence | Comment
---|---|---|---|---
4 | 32 | $\\{[\pm 4]^{1},[\pm 2\sqrt{3}]^{4},[0]^{22}\\}$ | $-$ | [8]
8 | 64 | $\\{[\pm 8]^{1},[\pm 4\sqrt{3}]^{4},[0]^{54}\\}$ | ? |
8 | 256 | $\\{[\pm 8]^{1},[\pm 4\sqrt{3}]^{20},[0]^{214}\\}$ | ? |
10 | 50 | $\\{[\pm 10]^{1},[\pm 5\sqrt{3}]^{2},[0]^{44}\\}$ | $-$ | $q\not\in\mathbb{Z}$
10 | 200 | $\\{[\pm 10]^{1},[\pm 5\sqrt{3}]^{12},[0]^{174}\\}$ | $-$ | $q\not\in\mathbb{Z}$
10 | 500 | $\\{[\pm 10]^{1},[\pm 5\sqrt{3}]^{32},[0]^{434}\\}$ | ? |
12 | 96 | $\\{[\pm 12]^{1},[\pm 6\sqrt{3}]^{4},[0]^{86}\\}$ | ? |
12 | 384 | $\\{[\pm 12]^{1},[\pm 6\sqrt{3}]^{20},[0]^{342}\\}$ | ? |
14 | 196 | $\\{[\pm 14]^{1},[\pm 7\sqrt{3}]^{8},[0]^{178}\\}$ | ? |
14 | 1372 | $\\{[\pm 14]^{1},[\pm 7\sqrt{3}]^{64},[0]^{1242}\\}$ | ? |
16 | 128 | $\\{[\pm 16]^{1},[\pm 8\sqrt{3}]^{4},[0]^{118}\\}$ | ? |
16 | 512 | $\\{[\pm 16]^{1},[\pm 8\sqrt{3}]^{20},[0]^{470}\\}$ | ? |
16 | 2048 | $\\{[\pm 16]^{1},[\pm 8\sqrt{3}]^{84},[0]^{1878}\\}$ | ? |
20 | 100 | $\\{[\pm 20]^{1},[\pm 10\sqrt{3}]^{2},[0]^{94}\\}$ | ? |
20 | 160 | $\\{[\pm 20]^{1},[\pm 10\sqrt{3}]^{4},[0]^{150}\\}$ | ? |
20 | 250 | $\\{[\pm 20]^{1},[\pm 10\sqrt{3}]^{7},[0]^{234}\\}$ | ? |
20 | 400 | $\\{[\pm 20]^{1},[\pm 10\sqrt{3}]^{12},[0]^{374}\\}$ | ? |
20 | 640 | $\\{[\pm 20]^{1},[\pm 10\sqrt{3}]^{20},[0]^{598}\\}$ | ? |
20 | 1000 | $\\{[\pm 20]^{1},[\pm 10\sqrt{3}]^{32},[0]^{934}\\}$ | ? |
20 | 1600 | $\\{[\pm 20]^{1},[\pm 10\sqrt{3}]^{52},[0]^{1494}\\}$ | ? |
20 | 2500 | $\\{[\pm 20]^{1},[\pm 10\sqrt{3}]^{82},[0]^{2334}\\}$ | ? |
20 | 4000 | $\\{[\pm 20]^{1},[\pm 10\sqrt{3}]^{132},[0]^{3734}\\}$ | ? |
$\vdots$ | $\vdots$ |  |  |
Table 4: Feasible periodic graphs whose $A$-spectra are the form $\\{[\pm
k]^{1},[\pm\frac{\sqrt{3}}{2}k]^{a},[0]^{b}\\}$.

##  6 Discussion and questions

In this paper, we completely determined bipartite regular graphs with four
distinct adjacency eigenvalues that induce periodic Grover walks, and show
that it is only $C_{6}$. In addition, we showed that there are only three
kinds of the second largest eigenvalues of bipartite regular periodic graphs
with five distinct eigenvalues. Focusing on certain conditions obtained from
walk-regularity and the number of quadrangles, we have narrowed down periodic
graphs. However, there are still many in the tables whose existence is
unknown. Table 2 is directly related to the existence and classification of
integral graphs, which is a problem in spectral graph theory. Indeed, the case
$k=4$ in Table 2 was completed by the results of Stevanović [33]. Similar
problems for the case $k\geq 6$ remain to be solved:

######  Question 6.1.

Fix an even number $k\geq 6$. Classify the integral graphs whose $A$-spectra
are the form $\\{[\pm k]^{1},[\pm\frac{k}{2}]^{a},[0]^{b}\\}$.

None of the graphs in Table 4 have been found so far. The existence of such
graphs should also be investigated:

######  Question 6.2.

Is there a graph whose A-spectrum is the form $\\{[\pm
k]^{1},[\pm\frac{\sqrt{3}}{2}k]^{a},[0]^{b}\\}$?

## Acknowledgements

S.K. is supported by JSPS KAKENHI (Grant No. 20J01175).

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EPJ Web of Conferences INPC 2013

11institutetext: Laboratoire de Physique Corpusculaire de Caen, ENSICAEN,
UniversitÃ de Caen, CNRS-IN2P3, Boulevard MarÃchal Juin, 14050 Caen, France

# Nuclear stopping for heavy-ion induced reactions in the Fermi Energy range :
from 1-Body to 2-Body dissipation

Olivier Lopez
(INDRA collaboration) 11<EMAIL_ADDRESS>
###### Abstract

We address the stopping in heavy-ion induced reactions around the Fermi energy
using central collisions recorded with _INDRA_ $4\pi$ array. The stopping is
minimal around the Fermi energy and corresponds to the disappearance of the
Mean-Field effects and the appearance of nucleon-nucleon elastic collisions.
This phenomenon is attributed to a change in the dissipation regime going from
1-body (Mean-Field) dissipation at low incident energy to 2- body (nucleonic
collisions) above the Fermi energy. A connection to the in-medium transport
properties of nuclear matter is proposed and comprehensive values of the
nucleon mean free path $\lambda_{NN}$ and the nucleon-nucleon cross section
are extracted.

##  1 Introduction

The study of transport phenomena is of utmost importance for understanding the
fundamental properties of nuclear matterDurand2006 . They are also critical in
the description of the supernova collapse and the formation of a neutron
starLattimer2004 . Transport properties of nuclear matter are also one of the
basic ingredients for microscopic modelsAndronic2006 . In this study, we are
looking at the global energy dissipation achieved in heavy- ion induced
reactions in the Fermi energy domain. We are using the large dataset available
in this energy range for symmetric systems recorded with the $4\pi$ array
_INDRA_Pouthas1995 . We are looking at central collisions, emphi.e.
corresponding to the maximal overlap between the 2 incoming nuclei. Doing so,
we can extract information concerning the stopping encountered in such
collisions and relate it tentatively to the nucleon mean free path in the
nuclear medium as well as the in-medium quenching factor for the nucleon-
nucleon cross section.

##  2 Stopping for central collisions

We are in this work focused on the _INDRA_ data coming from central
collisions, selected through the total multiplicity of charged particle as
done in Lehaut2010 . The data span a large body of symmetric systems, with
total mass between 70 and 500, and incident energies covering the full range
of the Fermi domain $E_{inc}/A=10-100\leavevmode\nobreak\ MeV$. To evaluate
the degree of stopping achieved in central collisions, we are using the energy
isotropy ratio $R_{E}$ defined as :

$R_{E}=\frac{1}{2}\frac{\sum_{i}E_{i}^{\perp}}{\sum_{i}E_{i}^{//}}$ (1)

The sum $i$ runs over the total number of charged particle in the event,
$E_{i}{\perp}=E_{i}\sin^{2}(\theta_{i})$ and $E_{i}{\
perp}=E_{i}\cos^{2}(\theta_{i})$ are respectively the transverse and parallel
energies in the center-of-mass of the reaction, $E_{i}$ and $\theta_{i}$ being
the total $CM$ energy and polar angle. By construction, this ratio is taking
values between $0$ and $1$, $1$ corresponding to an isotropic emission. We
present in figure 1 the mean isotropy ratio $<R_{E}>$ calculated for all
systems as a function of the incident energy.

Figure 1: Isotropy ratio $R_{E}$ as a function of incident energy for central
collisions in symmetric systems.

The coloured curves on figure 1 correspond to several analytical calculations
associated to the entrance channel; we consider here 2 Fermi spheres separated
by the relative momentum $\alpha P_{rel}$ corresponding to the relative energy
of the entrance channel. A quite straightforward calculation shows that the
energy isotropy ratio $R_{E}^{EC}(\alpha)$ is defined as :

$R_{E}^{EC(\alpha)}=\frac{1}{1+5(\frac{\alpha P_{rel}}{P_{Fermi}})^{2}}$ (2)

where $P_{Fermi}$ is the Fermi momentum at saturation density
($P_{Fermi}=266\leavevmode\nobreak\ MeV/c$), $P_{rel}=\mu V_{rel}$ is the
relative momentum, $\mu$ is the reduced mass and $V_{rel}$ the relative
velocity. $\alpha$ is a free parameter which represents the degree of
dissipation observed in the collision; $\alpha=0$ means full dissipation while
$\alpha=1$ corresponds to the sudden approximation with no dissipation. The
data are then dispatched between these two extreme values as shown in Fig. 1.
We note that the closest distance to the _no dissipation_ scenario ($\
alpha=1$) corresponds to an incident energy slightly below the Fermi energy,
here around $30A\leavevmode\nobreak\ MeV$. To better quantify this effect, we
compute the distance
$d=\frac{R_{E}-R_{E}^{EC}(\alpha=1}{R_{E}^{EC}(\alpha=0)-R_{E}^{EC}(\alpha=1)}$
between the data and the entrance channel, normalized to the distance between
the values corresponding to $\alpha=0$ and $\ alpha=1$. The results are
displayed in figure 2. A minimum around $E_{inc}/A=30\leavevmode\nobreak\ MeV$
is clearly visible, altogether with the decrease from barrier energy to Fermi
energy and the increase above the Fermi energy. This is attributed to the
change in the energy dissipation regime as already explained in Lehaut2010 .

Figure 2: Normalized distance $d$ as a function of incident energy.

Above the Fermi energy, in the nucleonic regime, we also observe a nice mass
hierarchy concerning the stopping: the heavier the system, the larger the
stopping is. This is fully consistent with a _Glauber_ picture where the
degree of stopping is closely related to the number of participants. To go
further, we can normalize the isotropy ratio to the characteristic size of the
system, here the radius $L=r_{0}A^{1/3}$ with $r_{0}=1.2\leavevmode\nobreak\
fm$. Figure 3 displays the results for this reduced quantity.

Figure 3: Reduced isotropy $R_{E}^{\prime}=R_{E}/L$ as a function of incident
energy. See text for details.

We observe that all systems seem to collapse to the same value at high energy,
well above the Fermi energy, above $100A\leavevmode\nobreak\ MeV$. This is
here an indication that the stopping is related to the characteristic size $L$
of the system, and is then connected to the nucleon mean free path
$\lambda_{NN}$ in the nuclear medium; to be more specific, we have performed
simple Monte Carlo simulations of 2 Fermi spheres using the _Glauber_ approach
to check this assumption. We have then obtained the following relationship
between the in-medium mean free path $\lambda_{NN}$ and the normalized
distance $d$ :

$\lambda_{NN}\approx\frac{L}{d^{2/3}}$ (3)

##  3 Nucleon Mean Free Path in the medium

Applying the findings of the previous section concerning $\lambda_{NN}$, we
obtain the results displayed by Fig. 4. It is interesting to note that all
curves exhibit the same behaviour whatever the total mass. We observe a
maximum at incident energy corresponding to minimal stopping, here at
$E_{inc}=30A\leavevmode\nobreak\ MeV$. The decrease at low incident energy
(below $30A\leavevmode\nobreak\ MeV$), can be attributed to the fact that the
Mean-Field dissipation is well active in this energy domain, and consequently
the normalized distance from the entrance channel is not anymore correct; in
this domain, we should adjust the $\alpha$ parameter to describe at best the
energy dissipation. In the limited scope of the present study, we are mainly
interested on the nucleonic regime. Going back to this incident energy domain,
we observe a decrease from $\lambda_{NN}\approx 8\leavevmode\nobreak\ fm$ to
$\lambda_{NN}=5\leavevmode\nobreak\ fm$. It is nice to see that the value at
high energy is in agreement with recent _Dirac-Brueckner-Hartree-Fock_
calculations using realistic interactions Rios2012 .

Figure 4: Nucleon Mean Free Path $\lambda_{NN}$ as a function of incident
energy.

##  4 In-medium effects for nucleon-nucleon cross section

Having the mean free path, we can now estimate the nucleon-nucleon cross
section by taking the usual formula from the kinetic theory of gases :
$\sigma_{NN}=1/(\rho\lambda_{NN})$ where $\rho$ is the nuclear density, taken
as the saturation density $\rho=0.16\leavevmode\nobreak\ fm^{-3}$. To get the
_true_ nucleon-nucleon cross section, we have to correct from the _Pauli_
blocking as a trivial (but essential) in-medium effect. To do so, we use the
prescription of ref. citeKikuchi1968. More sophisticated approaches concerning
the _Pauli_ blocking estimation can be found in the litterature Chen2013 , but
we consider here that the estimated correction factor is correct. We then
construct the quenching factor $F=\sigma_{NN}^{in-medium}/\sigma_{NN}^{free}$
by taking the energy-dependent free nucleon-nucleon cross section
$\sigma_{NN}^{free}$ available in the litterature Metropolis1958 .

Figure 5: In-medium correction factor $F$ for nucleon-nucleon cross section as
a function of incident energy.

The results are visible on Fig. 5 for the quenching factor $F$. We display
results only for the energy range above $E_{inc}/A=30\leavevmode\nobreak\
MeV$. The ’experimental’ values are compared to standard theoretical
prescriptions derived from different works and currently used in microscopic
transport models Klakow1993 ; LiMachleidt1993 ; Schnell1998 ; Xiangzhou1998 ;
Coupland2011 . From the experimental results, we see that the medium effects
are quenching almost all nucleon-nucleon collisions at low energy with
$F\approx 20\%$. We obtain values close to $F=80\%$ at the highest incident
energy, here at $100A\leavevmode\nobreak\ MeV$. It is interesting to note that
all parametrizations, except the one of ref. LiMachleidt1993 , are converging
toward this value, indicating that the in-medium effects are correctly taken
into account around $80A-100A\leavevmode\nobreak\ MeV$ incident energy. For
the low energy domain, the parametrizations give quite different results. The
only describing correctly the experimental data is the one of ref.
Coupland2011 . This parametrizatio n is based upon a phenomenological
assumption made by _Danielewicz_ concerning the quenching factor :

$F=\frac{\sigma_{0}}{\sigma_{NN}^{free}}\tanh(\sigma_{NN}^{free}/\sigma_{0})$
(4)

where $\sigma_{0}$ is the minimal nucleon-nucleon cross section and is
geometrically related to the inter-particle distance in the nuclear medium,
such as : $\sigma_{0}=0.85/rho_{0}^{2/3}$. It is worthwhile to mention that
$\sigma_{0}$ uses the same functional dependence $\rho^{2/3}$ as the one
deduced from the _Glauber_ -type of Monte Carlo simulations.

##  5 Conclusions

In this study, we have estimated the nucleon mean free path and the related
nucleon-nucleon cross section in the nuclear medium from the degree of
stopping achieved in central collisions in the Fermi domain between $30A$ and
$100A\leavevmode\nobreak\ MeV$. The estimated values for the mean free path
are comprised between $8fm$ at $E_{inc}/A=30\leavevmode\nobreak\ MeV$ and
$5fm$ at $E_{inc}/A=100\leavevmode\nobreak\ MeV$. The relatively large value
around the Fermi energy suggests that full thermalization cannot be reached in
such central collisions. These values are moreover in agreement with recent
theoretical findings using microscopic approaches. In-medium effects, _Pauli_
blocking and _high-order correlations_ in medium, have also been
experimentally inferred and are found to be large in the Fermi energy range;
this is a clear indication that these effects have to be properly taken into
account in any microscopic transport model. Concerning the quenching factor in
nuclear medium, the best agreement is obtained by using the prescription of
Coupland2011 . These results have nevertheless to be confirmed in a
forthcoming analysis. Anyway, this analysis shows that the stopping can be
used as a powerful probe to study transport properties of nuclear matter.

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