# On bipartite distance-regular Cayley graphs with small diameter

Edwin R. van Dam  Department of Econometrics and O.R., Tilburg University, The
Netherlands<EMAIL_ADDRESS>and  Mojtaba Jazaeri  Department of
Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran  School of
Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box:
19395-5746, Tehran, Iran<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study bipartite distance-regular Cayley graphs with diameter three or four.
We give sufficient conditions under which a bipartite Cayley graph can be
constructed on the semidirect product of a group — the part of this bipartite
Cayley graph which contains the identity element — and $\mathbb{Z}_{2}$. We
apply this to the case of bipartite distance-regular Cayley graphs with
diameter three, and consider cases where the sufficient conditions are not
satisfied for some specific groups such as the dihedral group. We also extend
a result by Miklavič and Potočnik that relates difference sets to bipartite
distance-regular Cayley graphs with diameter three to the case of diameter
four. This new case involves certain partial geometric difference sets and —
in the antipodal case — relative difference sets.

###### Key words and phrases:

Cayley graph; Bipartite distance-regular graph; Symmetric design; Difference
set; Partial geometric difference set; Relative difference set

###### 2010 Mathematics Subject Classification:

05B10, 05E30

##  1\. Introduction

We study bipartite distance-regular Cayley graphs with diameter three or four.
The subjects of both distance-regular graphs and Cayley graphs form important
areas in algebraic graph theory. For background (and more) on distance-regular
graphs, we refer to the monograph [5], survey [14], and website [3]. The
question which distance-regular graphs are Cayley graphs is a problem which
has received increasing attention recently (see [14, problem 70]). Miklavič
and Potočnik [27] for example classified distance-regular Cayley graphs on
dihedral groups. It turned out that every non-trivial distance-regular Cayley
graphs on a dihedral group is bipartite with diameter three. This gives rise
to the question which bipartite distance-regular graphs with diameter three
are Cayley graphs. Is it the case that all such distance-regular Cayley graphs
can be realized on the semidirect product of a group and $\mathbb{Z}_{2}$? We
note that, in general, for a given bipartite graph, it is an NP-complete
problem to decide if it has an automorphism of order $2$ which interchanges
the two parts; see [2, p. 106].

In this paper, we give sufficient conditions under which a bipartite Cayley
graph can be constructed on the semidirect product of a group — the part of
this bipartite Cayley graph which contains the identity element — and
$\mathbb{Z}_{2}$. We apply this to the case of bipartite distance-regular
Cayley graphs with diameter three, and consider cases where the sufficient
conditions are not satisfied for some specific groups such as the dihedral
group.

The development of a difference set is well known to be a symmetric design. In
turn, the incidence graph of a symmetric design is a bipartite distance-
regular graph with diameter three. Miklavič and Potočnik [27] made precise
when this construction from a difference set leads to a Cayley graph. Here we
extend this to the case of diameter four. This new case involves certain
partial geometric difference sets and — in the antipodal case — relative
difference sets.

We note that Chen and Li [7] studied relative difference sets in relation to
antipodal distance-regular graphs with diameter three. In this case the
connection set of the Cayley graph is the relative difference set, just like
partial difference sets are connection sets of strongly regular graphs (i.e.,
distance-regular graphs with diameter two).

The paper is further organized as follows. In Section 2, we introduce notation
and the relation between distance-regular graphs with diameter three and
difference sets. We recall the above mentioned correspondence between
difference sets and Cayley graphs by Miklavič and Potočnik [27] in Proposition
2.1. We add to this an observation about constructing bipartite Cayley graphs
from sets in an abelian group in Proposition 2.2.

In Section 3, we consider bipartite Cayley graphs. We give conditions on the
order and size such that a bipartite Cayley graph can be constructed on the
semidirect product of a group — the part of this bipartite Cayley graph which
contains the identity element — and $\mathbb{Z}_{2}$. In Proposition 3.5, we
extend this result to the case when the graph has no eigenvalue $0$. Before
applying all this in Section 5, we first introduce certain partial geometric
difference sets in Section 4, as these are relevant both for diameters three
and four. In Proposition 4.1, we extend the result by Miklavič and Potočnik
[27] by relating certain partial geometric difference sets to bipartite
distance-regular Cayley graphs with diameter four. We apply this to some
examples before zooming in on the antipodal bipartite case in Proposition 4.2.
Also here we mention some interesting examples.

In the final section, we then apply the results of Section 3 to the case of
bipartite distance-regular Cayley graphs with diameter three, and consider
some specific groups such as the dihedral group.

##  2\. Distance-regular Cayley graphs and difference sets

###  2.1. Preliminaries

Let $\Gamma$ be a connected graph with diameter $d$. Then $\Gamma$ is called
distance-regular with intersection array $\\{b_{0},b_{1},\ldots,b_{d-1};\break
c_{1},c_{2},\ldots,c_{d}\\}$ whenever, for each pair of vertices $x$ and $y$
at distance $i$, where $i=0,1,\ldots,d$, the number of neighbours of $x$ at
distance $i+1$ and $i-1$ from $y$ are constant numbers $b_{i}$ and $c_{i}$,
respectively. This implies that a distance-regular graph is regular with
valency $b_{0}=k$ and that the number of vertices at distance $i$ from a fixed
vertex is constant. This number is denoted by $k_{i}$ and it follows that
$k_{i+1}=\frac{k_{i}b_{i}}{c_{i+1}}$, where $i=0,1,\ldots,d-1$. Also the
number of neighbours of $x$ at distance $i$ from $y$ is a constant number
$k-b_{i}-c_{i}$, which is denoted by $a_{i}$. If $\Gamma$ is bipartite, then
$a_{i}=0$ for $i=0,1,\ldots,d$.

Let $G$ be a finite group and $S$ be an inverse-closed subset of $G$ not
containing the identity element; we call $S$ the connection set. Then the
Cayley graph $\mathop{\rm Cay}\nolimits(G,S)$ is the graph whose vertex set is
$G$, where two vertices $a$ and $b$ are adjacent (denoted by $a\sim b$)
whenever $ab^{-1}\in S$. By $S_{i}$ we denote the set of elements at distance
$i$ in $\mathop{\rm Cay}\nolimits(G,S)$ from the identity element. Recall that
a graph $\Gamma$ is a Cayley graph if and only if there exists a subgroup of
the automorphism group of $\Gamma$ which acts regularly on the vertex set of
$\Gamma$ (see e.g., [4, Lemma 16.3]). In this paper, the identity element of a
group $G$ is denoted by $e$ and its order by $|G|$.

A semidirect product of a group $H$ with a group $K$, which is denoted by
$H\rtimes K$ or $K\ltimes H$, is a (not necessarily unique) group $G$
containing a normal subgroup $H_{1}$ which is isomorphic to $H$ and a subgroup
$K_{1}$ isomorphic to $K$ such that $G=H_{1}K_{1}$ and $H_{1}\cap
K_{1}=\\{e\\}$.

###  2.2. Symmetric designs and difference sets

A $2$-$(n,k,\mu)$ design consists of a finite set of order $n$ (of elements
called points) and a family of $k$-element subsets (with $k\geq 2$) of this
set (called blocks) such that each pair of points is included in exactly $\mu$
blocks. Moreover, this design is called symmetric whenever the numbers of
points and blocks are equal. The incidence graph of a symmetric
$2$-$(n,k,\mu)$ design is a bipartite graph with two parts of points and
blocks such that a point is adjacent to a block whenever the point lies in the
block. It is known that this incidence graph is a bipartite distance-regular
graph with diameter three, and the other way around.

Indeed, let $\Gamma$ be a bipartite distance-regular graph with diameter $3$.
Then its intersection array is $\\{k,k-1,k-\mu;1,\mu,k\\}$, where $\mu=c_{2}$;
see also the distance-distribution diagram below. It is well known that the
graph $\Gamma$ is the incidence graph of a symmetric $2$-$(n,k,\mu)$ design,
where $2n$ is the number of vertices of this graph (see e.g., [16, Thm.
5.10.3]). Note that if such a design exists, then $k(k-1)=(n-1)\mu$, for
example because $n-1=k_{2}=\frac{k(k-1)}{\mu}$. If $k<n-1$ (in which case we
call the graph non-trivial), then the distance-$3$ graph $\Gamma_{3}$ is also
a bipartite distance-regular graph with intersection array
$\\{n-k,n-k-1,k-\mu;1,n-2k+\mu,n-k\\}$, which is the incidence graph of the
so-called complementary symmetric $2$-$(n,n-k,n-2k+\mu)$ design. Furthermore,
$\\{k^{1},\sqrt{k-\mu}^{[n-1]},-\sqrt{k-\mu}^{[n-1]},-k^{1}\\}$ is the
spectrum (the multiset of all eigenvalues of the adjacency matrix) of this
graph.

Figure 1. Distance-distribution diagram

Trivial examples in the context of this paper are the complete bipartite graph
$K_{n,n}$ minus a perfect matching. These are the incidence graphs of the
(trivial) $2$-$(n,n-1,n-2)$ designs; or the bipartite distance-regular graphs
with intersection array $\\{k,k-1,1;1,k-1,k\\}$, i.e., those with $k=n-1$, or
equivalently $\mu=k-1$. Such graphs are Cayley graphs on the dihedral group
$D_{2n}$ (see e.g., [13, §3.1]).

An $(n,k,\mu)$-difference set in a finite group $H$, where $|H|=n$, is a
subset $D\subseteq H$ of size $k$ such that every non-identity element of $H$
can be expressed exactly $\mu$ times as (a “multiplicative difference”)
$d_{1}d_{2}^{-1}$, where $d_{1}$, $d_{2}$ are elements of $D$. Moreover, from
an $(n,k,\mu)$-difference set $D$ in a finite group $H$, one can construct a
symmetric $2$-$(n,k,\mu)$ design called the development of $D$, by considering
$\\{Dh\mid h\in H\\}$ as the set of blocks. We note that if $D$ is a
difference set, then $H\setminus D$ is also a difference set, and its
development is the complementary design of the development of $D$. Moreover,
$D^{-1}$ is also a difference set (even when $H$ is nonabelian), because its
development gives the dual design (i.e., the design obtained by interchanging
the role of points and blocks). An $(n,k,\mu)$-difference set $D$ is called
trivial whenever $k\in\\{0,1,n-1,n\\}$.

We recall the following result by Miklavič and Potočnik [27] that relates
bipartite distance-regular Cayley graphs with diameter $3$ and certain
difference sets.

######  Proposition 2.1.

[27, Lemma 2.8] Let $G$ be a group of order $2n$ and $S$ a subset of $G$ of
size $k$. Then the following statements are equivalent:111In the formulation
in [27] we replaced $D=a^{-1}S$ by $D=Sa^{-1}$ because of our slightly
different definition of Cayley graphs. The original formulation is also valid
with our definition, but it is less natural.

  * •

$S\subseteq G\setminus\\{e\\}$, $S=S^{-1}$ and $\mathop{\rm
Cay}\nolimits(G,S)$ is a non-trivial bipartite distance-regular graph with
diameter $3$ and intersection array $\\{k,k-1,k-\mu;1,\mu,k\\}$;

  * •

there is a subgroup $H$ of index $2$ in $G$ and an element $a\in G\setminus H$
such that the set $D=Sa^{-1}$ is a non-trivial $(n,k,\mu)$-difference set in
$H$ satisfying $D^{-1}=aDa$;

  * •

there is a subgroup $H$ of index $2$ in $G$ such that for every $a\in
G\setminus H$, the set $D=Sa^{-1}$ is a non-trivial $(n,k,\mu)$-difference set
in $H$ satisfying $D^{-1}=aDa$.

For abelian groups, we can be a bit more specific (and constructive).

######  Proposition 2.2.

Let $D$ be a subset (not necessarily a difference set) of an abelian group
$H$. Then the incidence graph of its development is isomorphic to the Cayley
graph on the generalized dihedral group $G=\mathop{\rm
Dih}\nolimits(H)=H\rtimes\mathbb{Z}_{2}$ with connection set $S=Dc$, where
$c^{2}=1$ ($c\notin H$) and $chc=h^{-1}$ for every element $h\in H$.

###### Proof.

We note that $S$ contains only elements of order $2$ and hence it is inverse-
closed. The map that sends (point) $h$ to $h$ and (block) $Dh$ to $h^{-1}c$ is
an isomorphism. Indeed, if $h_{1}\sim Dh_{2}$ (in the incidence graph), then
$h_{1}h_{2}^{-1}\in D$. It follows that
$h_{1}(h_{2}^{-1}c)^{-1}=h_{1}ch_{2}=h_{1}h_{2}^{-1}c\in Dc$ and therefore
$h_{1}$ is adjacent to $h_{2}^{-1}c$ in the Cayley graph, and the other way
around. ∎

This lemma applies for example in the case of the incidence graph of a
Desarguesian projective plane (as we observed before in [13, § 3.5]). Recall
that a projective plane of order $q$ is a symmetric $2$-$(q^{2}+q+1,q+1,1)$
design and the incidence graphs of projective planes are precisely the
bipartite distance-regular graphs with diameter three and girth $6$ (i.e.,
with $c_{2}=1$); these have intersection array $\\{k,k-1,k-1;1,1,k\\}$, with
$k=q+1$. Indeed, it is well known that a Desarguesian projective plane can be
constructed as the development of a (Singer) difference set in a cyclic group,
hence by the above lemma, its incidence graph is indeed a Cayley graph, more
specifically on a dihedral group. We conclude the following (well-known)
result.

######  Proposition 2.3.

Let $\Gamma$ be a bipartite distance-regular graph with diameter $3$ and girth
$6$. Then $\Gamma$ is the incidence graph of a projective plane. Moreover, if
the projective plane is Desarguesian, then $\Gamma$ is a Cayley graph on a
dihedral group.

For a table of abelian difference sets with small parameters, we refer to [19,
Table 18.73].

We finally note that there are non-abelian difference sets for which the
development is not a Cayley graph. Before we go deeper into distance-regular
Cayley graphs, we now first derive some elementary results on bipartite Cayley
graphs.

##  3\. Bipartite Cayley graphs

In this section, we will first consider some elementary properties of
bipartite Cayley graphs, mostly motived by Proposition 2.2. In Section 3.4, we
will go deeper into the case of nonsingular graphs, which will all be used in
Section 5 when we continue with the study of bipartite distance-regular Cayley
graphs with diameter $3$.

###  3.1. Normal subgroups and the semidirect product

######  Lemma 3.1.

Let $\mathop{\rm Cay}\nolimits(G,S)$ be a connected bipartite Cayley graph.
Then the part of this bipartite graph which contains the identity element is a
normal subgroup $H$ of index $2$ in the group $G$ and $S\subseteq G\setminus
H$.

###### Proof.

Let $H$ be the part of the bipartite graph $\mathop{\rm Cay}\nolimits(G,S)$
which contains the identity element $e$. If $a,b\in H$, then $ba\in H$. To see
this let $d(e,b)=2n$, where $n\in\mathbb{N}$. Then there exists a path of
length $2n$ between $a$ and $ba$ which implies that $ba\in H$. Moreover,
similarly, $a^{-1},b^{-1}\in H$. This implies that $H$ is a subgroup of $G$.
On the other hand, each part of this regular bipartite graph has the same size
and therefore the subgroup $H$ is normal since the index of this subgroup in
$G$ is $2$ and this completes the proof. ∎

######  Lemma 3.2.

Let $\Gamma=\mathop{\rm Cay}\nolimits(G,S)$ be a connected bipartite Cayley
graph of order $2n$ and valency $k$ and let $H$ be the part of this bipartite
graph which contains the identity element. If $n$ is odd or $k$ is odd, then
$G$ is isomorphic to $H\rtimes\mathbb{Z}_{2}$.

###### Proof.

First, let $n$ be an odd number. Then the normal subgroup $H$ has odd order
$n$. Because $G$ has even order, it contains an involution $a$, which clearly
cannot be in $H$. Therefore $G=H\rtimes\langle a\rangle$.

Secondly, let $k$ be an odd number, so $|S|$ is odd. Because $S=S^{-1}$, the
number of $s\in S$ that is not an involution (i.e., for which $s\neq s^{-1}$)
is even, so there must be an involution $a\in S$. Therefore $G=H\rtimes\langle
a\rangle$. ∎

###  3.2. Complete bipartite graphs

Let us first make a few observations about the bipartite distance-regular
graphs with diameter two: the regular complete bipartite graphs $K_{n,n}$,
with $n>1$. These are Cayley graphs for any group $G$ of order $2n$ having a
subgroup $H$ of order $n$, by considering $S=G\setminus H$. If moreover $S$
contains an involution $a$, then $G=H\rtimes\langle a\rangle$. For odd $n$, it
is clear from the above that this is the case. However, for even $n$, it is
different. For example if $G$ is the (abelian) group $G=\mathbb{Z}_{2n}$,
which has a (abelian) subgroup $H$ (of even numbers) isomorphic to
$\mathbb{Z}_{n}$, but $G\setminus H$ does not have an involution. Still,
$K_{n,n}$ can easily be constructed on a semidirect product
$H\rtimes\mathbb{Z}_{2}$ such as the dihedral group. We will also see this
distinction for bipartite distance-regular graphs with larger diameter.

###  3.3. No involutions

Let us try to find a general setting for the previous example, i.e., in the
case that there are no involutions in $G\setminus H$. Consider a bipartite
Cayley graph $\Gamma=\mathop{\rm Cay}\nolimits(G,S)$ where the usual subgroup
$H$ is such that a semidirect product of $H$ and $\mathbb{Z}_{2}$ can be
defined (such as when $H$ is abelian). For $a\in G\setminus H$, we let
$T_{a}=Sa^{-1}$, which is a subset of $H$. Let $G^{\prime}=H\rtimes\langle
c\rangle$, with $c^{2}=e$, but $c\notin G$. We now take $S^{\prime}=T_{a}c$.
If $S^{\prime}$ is inverse-closed (in $G^{\prime}$), then we can use it as a
connection set and obtain a bipartite Cayley graph
$\Gamma^{\prime}=\mathop{\rm Cay}\nolimits(G^{\prime},S^{\prime})$.

######  Lemma 3.3.

The Cayley graph $\Gamma^{\prime}=\mathop{\rm
Cay}\nolimits(G^{\prime},S^{\prime})$ is isomorphic to $\Gamma=\mathop{\rm
Cay}\nolimits(G,S)$.

###### Proof.

The map $\varphi:\mathop{\rm Cay}\nolimits(G,S)\rightarrow\mathop{\rm
Cay}\nolimits(G^{\prime},S^{\prime})$ defined by

$\varphi(h)=h,\varphi(a^{-1}h)=ch,$

for $h\in H$, is an isomorphism, as is easily checked. ∎

As a corollary, we obtain the following (cf. Proposition 2.2).

######  Corollary 3.4.

Let $\Gamma=\mathop{\rm Cay}\nolimits(G,S)$ be a connected bipartite Cayley
graph and let $H$ be the part of this bipartite graph which contains the
identity element. If $H$ is abelian, then $\Gamma$ can be constructed as a
Cayley graph on the generalized dihedral group $\mathop{\rm Dih}\nolimits(H)$.

###### Proof.

We can apply the above construction with $G^{\prime}=\mathop{\rm
Dih}\nolimits(H)$ for any $a\in G\setminus H$. The set $S^{\prime}$ is
inverse-closed because all elements in $\mathop{\rm Dih}\nolimits(H)\setminus
H$ are involutions. ∎

A particular example where this applies (and that is more interesting than the
complete bipartite graph) is the following description as a Cayley graph of
the $4$-cube (a bipartite distance-regular graph with diameter $4$); the only
$4$-regular bipartite Cayley graph on $16$ vertices with integral eigenvalues,
according to Minchenko and Wanless [28].

Let $G=\mathbb{Z}_{4}\times\mathbb{Z}_{4}=\langle a,b\mid
a^{4}=b^{4}=1,ab=ba\rangle$ and $S=\\{a,a^{-1},b,b^{-1}\\}$. Then $H=\langle
ab,a^{2}\rangle$, which is isomorphic to $\mathbb{Z}_{4}\times\mathbb{Z}_{2}$,
but $G\setminus H$ does not have involutions. However, the $4$-cube can also
be described on $\mathop{\rm Dih}\nolimits(H)$, or
$(\mathbb{Z}_{4}\times\mathbb{Z}_{2})\rtimes\mathbb{Z}_{2}=\langle
ab,a^{2},c\mid(ab)^{4}=(a^{2})^{2}=c^{2}=1,(ab)a^{2}=a^{2}(ab),(c(ab))^{2}=(ca^{2})^{2}=1\rangle$,
with connection set $S^{\prime}=Sa^{-1}c$.

We note that the above construction is not the only way to obtain isomorphic
bipartite Cayley graphs. For example, consider the below bipartite Cayley
graph on $18$ vertices and valency $4$. Note that this is the only such Cayley
graph with integral eigenvalues [28]; it is the bipartite double of the Paley
graph $P(9)$.

Indeed, let $G=\mathbb{Z}_{6}\times\mathbb{Z}_{3}=\langle a,b\mid
a^{6}=b^{3}=1,ab=ba\rangle$, $S=\\{a,a^{5},a^{3}b,a^{3}b^{2}\\}$,
$G^{\prime}=(\mathbb{Z}_{3}\times\mathbb{Z}_{3})\rtimes\mathbb{Z}_{2}=\langle
a^{2},b,c\mid(a^{2})^{3}=b^{3}=c^{2}=1,a^{2}b=ba^{2},ca^{2}c=a^{-2},cbc=b^{-1}\rangle$,
and $S^{\prime}=\\{c,a^{2}c,bc,a^{2}bc\\}$. The normal subgroup $H$ in $G$ is
isomorphic to $\mathbb{Z}_{3}\times\mathbb{Z}_{3}$ but it can be checked that
there is no element $g\in G\setminus H$, such that $S^{\prime}=Sg^{-1}c$.
Still, the corresponding Cayley graphs are isomorphic.

###  3.4. Bipartite nonsingular Cayley graphs

In this section, we consider bipartite Cayley graphs that have no eigenvalues
$0$, in order to obtain similar results as in the previous section for the
cases that $k$ is even and $n$ is even, but not divisible by $4$. Note that
bipartite distance-regular graphs with odd diameter have no eigenvalue $0$
(contrary to those with even diameter).

######  Proposition 3.5.

Let $\Gamma=\mathop{\rm Cay}\nolimits(G,S)$ be a connected bipartite Cayley
graph of order $2n$ and valency $k$ and let $H$ be the part of this bipartite
graph which contains the identity element. If $n$ is not divisible by $4$ and
$\Gamma$ has no eigenvalue $0$, then $G$ is isomorphic to
$H\rtimes\mathbb{Z}_{2}$.

###### Proof.

By Lemma 3.2, we may assume that $n$ is even. Assume that $n$ is not divisible
by $4$, so that $n=2m$ for some odd $m$. Then the Sylow $2$-subgroup $L$, say,
of $G$ has order $4$ and $L$ is not contained in the normal subgroup $H$ (the
latter which has order $n$).

If $L$ is cyclic, then we may use a so-called normal $p$-complement theorem.
Indeed, by [32, Cor. 10.24], there exists a normal subgroup $N$ of (odd) order
$m$ in the group $G$ such that $G=NL$ and $N\cap L=\\{e\\}$. Now the quotient
group $G/N$ is cyclic because $L$ is cyclic, so $G/N=\\{N,Na,Na^{2},Na^{3}\\}$
for some $a\notin N$. Note also that $N$ is contained in $H$, for otherwise
$G=H\cup gH$ for some $g\in N\setminus H$, so $G=NH$, which implies that
$|G|=8|H\cap N|$, which is a contradiction. Because we may assume that the
involution $a^{2}$ is in $H$, it follows that $H=N\cup Na^{2}$.

Next, we will use that the cosets of a normal subgroup form an equitable
partition in a Cayley graph [13, § 2.3]. In this case, we obtain an equitable
partition with four parts $G/N$ for $\Gamma$. Moreover, we have a very
particular quotient matrix. If $xa\in S$, where $x\in N$, then $(xa)^{-1}\in
Na^{3}$. This implies that the numbers of adjacent vertices to a fixed vertex
($x$) of $N$ in $Na$ and in $Na^{3}$ are equal. It follows that the quotient
matrix equals $k/2$ times the adjacency matrix of the $4$-cycle, which has
eigenvalue $0$. Thus, also $\Gamma$ has eigenvalue $0$ [6, Lemma 2.3.1], which
contradicts our assumption. Hence the Sylow $2$-subgroup $L$ cannot be cyclic.

Thus, $L$ is isomorphic to $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, and
$L\setminus H$ (which is nonempty) contains an involution $a$, and it follows
that $G=H\rtimes\langle a\rangle$. ∎

##  4\. Partial geometric difference sets

Before we apply the results of Section 3, we first need to introduce partial
geometric designs and difference sets. Particular cases of these are also
defined, as they naturally connect to bipartite distance-regular graphs with
diameter $4$.

###  4.1. Partial geometric designs and distance-regular graphs with diameter
$4$

A symmetric partial geometric design — or symmetric $1\frac{1}{2}$-design —
with parameters $(n,k,\alpha,\beta)$ is a $1$-design with $n$ points and $n$
blocks of size $k$ with the property that for each point-block pair $(p,B)$,
the number of incident point-block pairs $(p^{\prime},B^{\prime})$,
$p^{\prime}\neq p$, $B^{\prime}\neq B$, with $p^{\prime}\in B$ and $p\in
B^{\prime}$ equals $\beta$ or $\alpha$, depending on whether $p$ is in $B$ or
not, respectively. Note that every symmetric design is also a partial
geometric design.

The incidence graphs of symmetric partial geometric designs are precisely the
regular bipartite graphs with four or five distinct eigenvalues [15]. Not all
regular bipartite graphs with five distinct eigenvalues are distance-regular,
as in the case of four eigenvalues. It was shown however [9, Prop. 3.5] that
if in such a graph, the number of common neighbors of two vertices at distance
$2$ is constant ($c_{2}=\mu$), then it is distance-regular (with diameter
$4$).

On the other hand, it is known (see [5, Prop. 1.7.1]) that a bipartite
distance-regular graphs with diameter $4$ is the incidence graph of an
incidence structure called (square) partial $\lambda$-geometry, as introduced
by Drake [10]. We need not further define these but instead build on the
definition of (symmetric) partial geometric design with the additional
property that any two points meet in either $0$ or $\mu$ blocks, and dually,
any two blocks share either $0$ or $\mu$ points.222We denote the $\lambda$ in
partial $\lambda$-geometry by $\mu$.

In general, a bipartite distance-regular graph $\Gamma$ with diameter $4$ on
$2n$ vertices has intersection array
$\\{k,k-1,k-\mu,k-c_{3};1,\mu,c_{3},k\\}$, so $\mu=c_{2}$, and
$c_{3}=\frac{k(k-1)(k-\mu)}{(n-k)\mu}$ (which follows from the standards
relations between parameters). The halved graphs of $\Gamma$ are strongly
regular graphs, with parameters following from the above intersection array.
Recall that if $\Gamma$ is a Cayley graph, then also these halved graphs are
Cayley graphs, which may give extra restrictions for existence as a Cayley
graph.

An example where this applies is the distance-regular graph on $100$ vertices
(and a related partial $5$-geometry on 50 points) that can be constructed from
the cocliques in the Hoffman-Singleton graph (see [5, Thm. 13.1.1]). This
graph is not a Cayley graph because its halved graphs are the complement of
the Hoffman-Singleton graph, which is known not to be a Cayley graph [25].

Note finally that if we view the (distance-regular) $\Gamma$ as the incidence
graph of a symmetric partial geometric design with parameters
$(n,k,\alpha,\beta)$, then it follows that $\alpha=\mu c_{3}$ and
$\beta=(k-1)(\mu-1)$ (we omit a derivation, which is similar as the later
derivation in the group case in Proposition 4.1).

###  4.2. Partial $\mu$-geometric difference sets

A partial geometric difference set — or $1\frac{1}{2}$-difference set — in a
finite group $H$ with parameters $(n,k,\alpha,\beta)$, as introduced by Olmez
[30], is a $k$-subset $D$ of $H$, where $|H|=n$, with the property that every
$h\in H$ can be expressed as $d_{1}d_{2}^{-1}d_{3}$, with
$d_{1},d_{2},d_{3}\in D$, in either $2k-1+\beta$ or $\alpha$ ways, depending
on whether $h\in D$ or not, respectively. Note that the contribution $2k-1$
comes from the “trivial” ways to express $h\in D$ as a required triple
product. As in the case of the usual difference sets, also here the
development of a partial geometric difference set $D$ is a symmetric partial
geometric design; moreover, $D^{-1}$ is also a partial geometric difference
set and its development is the dual design (which has the same parameters).

Given the above characterization of bipartite distance-regular graphs with
diameter $4$ among the incidence graphs of symmetric partial geometric
designs, it follows easily that the incidence graph of the development of a
partial geometric difference set $D$ is distance-regular with $c_{2}=\mu$ if
and only if every $h\in H\setminus\\{e\\}$ can be written as
$d_{1}d_{2}^{-1}$, with $d_{1},d_{2}\in D$ in $\mu$ or $0$ ways and — dually —
every $h\in H\setminus\\{e\\}$ can be written as $d_{1}^{-1}d_{2}$, with
$d_{1},d_{2}\in D$ in $\mu$ or $0$ ways. We call such a partial geometric
difference set a partial $\mu$-geometric difference set. Note that a
difference set is a degenerate case of this; in which case the diameter is $3$
instead of $4$, as we saw before.

We can now obtain a similar characterization as in Proposition 2.1.

######  Proposition 4.1.

Let $G$ be a group of order $2n$ and $S$ a subset of $G$ of size $k$. Then the
following statements are equivalent:

  1. (1)

$S\subseteq G\setminus\\{e\\}$, $S=S^{-1}$ and $\mathop{\rm
Cay}\nolimits(G,S)$ is a bipartite distance-regular graph with diameter $4$
and intersection array $\\{k,k-1,k-\mu,k-c_{3};1,\mu,c_{3},k\\}$;

  2. (2)

there is a subgroup $H$ of index $2$ in $G$ such that for every $a\in
G\setminus H$, the set $D=Sa^{-1}$ is a partial $\mu$-geometric difference set
with parameters $(n,k,\mu c_{3},(k-1)(\mu-1))$ satisfying $D^{-1}=aDa$;

  3. (3)

there is a subgroup $H$ of index $2$ in $G$ and an element $a\in G\setminus H$
such that the set $D=Sa^{-1}$ is a partial $\mu$-geometric difference set with
parameters $(n,k,\mu c_{3},(k-1)(\mu-1))$ satisfying $D^{-1}=aDa$.

###### Proof.

$(1)\Rightarrow(2)$: Assume (1) and let $H$ be the part of the bipartite graph
containing $e$. Let $a\in G\setminus H$ and $D=Sa^{-1}$. Then $D^{-1}=aDa$ and
(in group ring notation) we have that
$D^{-1}D=aS^{-1}Sa^{-1}=aS^{2}a^{-1}=a(k\\{e\\}+\mu S_{2})a^{-1}=k\\{e\\}+\mu
aS_{2}a^{-1}$, so every non-identity element of $H$ can be written as
$d_{1}d_{2}^{-1}$ in $\mu$ or $0$ ways. Dually the same holds because
$DD^{-1}=S^{2}=k\\{e\\}+\mu S_{2}$. Finally, $DD^{-1}D=kD+\mu S_{2}D=kD+\mu
S_{2}Sa^{-1}=kD+\mu((k-1)S+c_{3}S_{3})a^{-1}=kD+\mu(k-1)D+\mu c_{3}(H\setminus
D)=(2k-1)D+(\mu-1)(k-1)D+\mu c_{3}(H\setminus D),$ hence $D$ is a partial
$\mu$-geometric difference set with parameters $(n,k,\mu c_{3},(k-1)(\mu-1))$.

$(2)\Rightarrow(3)$: Trivial.

$(3)\Rightarrow(1)$: Assume (3), then $S$ is inverse-closed because
$D^{-1}=aDa$. From above considerations, it follows that the incidence graph
of the development of $D$ is a bipartite distance-regular graph $\Gamma$ with
diameter $4$ and intersection array $\\{k,k-1,k-\mu,k-c_{3};1,\mu,c_{3},k\\}$.
This incidence graph is isomorphic to $\mathop{\rm Cay}\nolimits(G,S)$. To see
this, consider the map $\varphi:\mathop{\rm
Cay}\nolimits(G,S)\rightarrow\Gamma$ with $\varphi(h)=h$ and
$\varphi(a^{-1}h)=Dh$, for $h\in H$. It is clear that $\varphi$ is a
bijection. Moreover, if $h_{1}\sim a^{-1}h_{2}$, then
$h_{1}(a^{-1}h_{2})^{-1}\in S=Da$, hence $h_{1}h_{2}^{-1}\in D$, $h_{1}\in
Dh_{2}$, and finally $\varphi(h_{1})\sim\varphi(a^{-1}h_{2})$, and the other
way around. ∎

For example, the Van Lint-Schrijver partial geometry $pg(5,5,2)$ [22] can be
constructed as the development of a partial $1$-geometric difference set on
the additive group of $GF(81)$. Indeed, if $\gamma$ is a primitive fifth root
of unity, then $D=\\{0,1,\gamma,\gamma^{2},\gamma^{3},\gamma^{4}\\}$ is such a
partial $1$-geometric difference set. Another nice description in the group
$\mathbb{Z}_{3}^{4}$ is also available [22, Construction 2]. Hence, by
Proposition 2.2, its incidence graph — the distance-regular Van Lint-Schrijver
graph — is a Cayley graph. The recently constructed other partial geometry
$pg(5,5,2)$ [8, 20] does not have a group that acts transitively on the
points, so its incidence graph is not a Cayley graph.

Also a few other examples are well known to be Cayley graphs, such as the
folded $8$-cube and folded $9$-cube.

Another example that is not a Cayley graph (next to the earlier mentioned
graph related to the Hoffman-Singleton graph) is the Leonard graph. This
follows because the corresponding design is not self-dual, and hence the
Leonard graph is not vertex-transitive [5, Thm. 11.4.4].

We finally note that in our previous paper [13, Prop. 3.3], we obtained the
condition that $s$ must be $0$ or $4\mod 6$ for the incidence graph of a
generalized quadrangle of order $s$ to be a Cayley graph. Also for $s=4$, it
is not a Cayley graph [13, Prop. 3.7].

###  4.3. Symmetric relative difference sets and antipodal graphs

A special case of partial $\mu$-geometric designs is the family of symmetric
transversal designs $STD_{\mu}[r\mu;r]$, or symmetric $(r,\mu)$-nets. Their
incidence graphs are precisely the antipodal bipartite distance-regular graphs
with intersection array $\\{r\mu,r\mu-1,(r-1)\mu,1;1,\mu,r\mu-1,r\mu\\}$. The
corresponding concept of partial $\mu$-geometric difference sets is that of
symmetric $(r\mu,r,r\mu,\mu)$-relative difference sets.

Let $H$ be a finite group and $N$ a proper subgroup of $H$ such that $|N|=r$
and $[H:N]=m$. Then a $k$-subset $D$ of $H$ is an $(m,r,k,\mu)$-relative
difference set relative to $N$ (the “forbidden” subgroup) whenever every $h\in
H\setminus\\{e\\}$ can be written as $d_{1}d_{2}^{-1}$, with $d_{1},d_{2}\in
D$ in $0$ or $\mu$ ways, depending on whether $h\in N$ or not, respectively.
Moreover, we say that $D$ is symmetric whenever $D^{-1}$ is also a relative
difference set (possibly with a different forbidden subgroup). Jungnickel [18]
showed that if $N$ is normal, then $D$ is symmetric.

Similar as before, the development of a symmetric $(r\mu,r,r\mu,\mu)$-relative
difference set is a symmetric transversal design $STD_{\mu}[r\mu;r]$, and
hence its incidence graph is an antipodal bipartite distance-regular graph
with diameter $4$ (in fact, it is an $r$-cover of a complete multipartite
graph).

The following is in some sense a special case of Proposition 4.1.

######  Proposition 4.2.

Let $G$ be a group of order $2r^{2}\mu$ and $S$ a subset of $G$. Then the
following statements are equivalent:

  1. (1)

$S\subseteq G\setminus\\{e\\}$, $S=S^{-1}$ and $\mathop{\rm
Cay}\nolimits(G,S)$ is an antipodal bipartite distance-regular graph with
diameter $4$ and intersection array
$\\{r\mu,r\mu-1,(r-1)\mu,1;1,\mu,r\mu-1,r\mu\\}$;

  2. (2)

there is a subgroup $H$ of index $2$ in $G$ and a subgroup $N$ of $H$ of order
$r$ such that for every $a\in G\setminus H$, the set $D=Sa^{-1}$ is a
symmetric $(r\mu,r,r\mu,\mu)$-relative difference set relative to $N$ in $H$
satisfying $D^{-1}=aDa$;

  3. (3)

there is a subgroup $H$ of index $2$ in $G$, a subgroup $N$ of $H$ of order
$r$, and an element $a\in G\setminus H$ such that the set $D=Sa^{-1}$ is a
symmetric $(r\mu,r,r\mu,\mu)$-relative difference set relative to $N$ in $H$
satisfying $D^{-1}=aDa$.

###### Proof.

Most of the proof is omitted because it is similar as the proof of Proposition
4.1. Additional details are as follows.

$(1)\Rightarrow(2)$: Let $N=S_{4}\cup\\{e\\}$. Because the Cayley graph is
antipodal, it follows that $N$ is a subgroup of $H$. It follows that
$D=Sa^{-1}$ is an $(r\mu,r,r\mu,\mu)$-relative difference set relative to $N$
in $H$. Similarly, $D^{-1}$ is an $(r\mu,r,r\mu,\mu)$-relative difference set
relative to the subgroup $a^{-1}Na$ in $H$. ∎

The antipodal bipartite distance-regular graphs with diameter $4$ and $\mu=1$
are incidence graphs of affine planes minus a parallel class of lines. As we
already observed before (but provided with few arguments) in [13], relative
difference sets in the abelian group $GF(q)^{2}$ are known for the
Desarguesian affine planes $AG(2,q)$ minus a parallel class of lines, for
prime powers $q$. Again, by Proposition 2.2, their incidence graphs, which are
antipodal bipartite distance-regular graphs with diameter $4$, are therefore
Cayley graphs.

Many more constructions of relative difference sets are known that give rise
to distance-regular Cayley graphs. For such constructions, we refer to the
survey paper on relative difference sets by Pott [31], for example.

A distinguished case is the one with $r=2$: symmetric transversal designs
$STD_{\mu}[2\mu;2]$ give rise to the Hadamard graphs (on $8\mu$ vertices),
which in turn are equivalent to (distance-regular) Hadamard matrices of size
$2\mu$; see [5, §1.8]. The smallest Hadamard graphs are $C_{8}$ and the
$4$-cube, which are clearly Cayley graphs. For $\mu=4$ and $\mu=6$, the
(unique) Hadamard graphs are also Cayley graphs, as they can be obtained from
appropriate relative difference sets (as in [33]).

We mention once more the distance-regular $4$-cube (recall also the earlier
remarks in Section 3), for it has a cospectral graph: the Hoffman graph. This
graph is neither a Cayley graph nor is it distance-regular. Still, it can be
constructed from a square transversal design; however its dual is not a
transversal design. In this case this implies that the Hoffman graph is only
“half distance-regular”. Likewise, Hiramine [17] constructed non-symmetric
relative difference sets, among others one with parameters $(12,3,12,4)$. It
gives rise to a half distance-regular graph that is cospectral to the Suetake
graph (the unique distance-regular graph with intersection array
$\\{12,11,8,1;1,4,11,12\\}$). The latter comes from the unique symmetric
transversal design $STD_{4}[12;3]$, see [34]. We found a related relative
difference set in the abelian group
$H=\mathbb{Z}_{2}\times\mathbb{Z}_{3}\times\mathbb{Z}_{6}$. Indeed, let

$D=\\{000,002,004,005,011,023,100,101,114,122,123,125\\}.$

######  Proposition 4.3.

The set $D$ is a relative difference set in the abelian group
$H=\mathbb{Z}_{2}\times\mathbb{Z}_{3}\times\mathbb{Z}_{6}$, relative to the
normal subgroup $N=\mathbb{Z}_{3}$. The incidence graph of its development is
the Suetake graph, which is therefore a Cayley graph.

Another very interesting antipodal bipartite distance-regular graph with
diameter $4$ is the hexacode graph on $36$ vertices [5, Thm. 13.2.2], which
comes from the unique $STD_{2}[6;3]$. We checked that its automorphism group
does not have a regular subgroup, so it is not a Cayley graph. Besides the
above two examples with $r=3$, we also mention the Pappus graph (the incidence
graph of the affine plane of order $3$ minus a parallel class of lines; see
above), and the four graphs coming from an $STD_{3}[9;3]$. The latter were
classified by Mavron and Tonchev [24]. We checked that only the one with the
smallest automorphism group is not a Cayley graph (in particular, it is not
even vertex-transitive). The distance-transitive one can be constructed from a
relative difference set in $\mathbb{Z}_{3}^{3}$, and the other two in
$\mathbb{Z}_{9}\times\mathbb{Z}_{3}$ and
$(\mathbb{Z}_{3}\times\mathbb{Z}_{3})\rtimes\mathbb{Z}_{3}$, respectively.

We finish this section by mentioning that we checked also that the distance-
transitive graph coming from an $STD_{2}[8;4]$ is a Cayley graph (in
particular, that it can be obtained from a relative difference set in
$\mathbb{Z}_{4}\times\mathbb{Z}_{4}\times\mathbb{Z}_{2}$).

##  5\. Bipartite distance-regular Cayley graphs with diameter $3$

We now return to bipartite distance-regular graphs with diameter three to go
deeper into the more general results in Section 3. The following result
follows immediately from there.

######  Proposition 5.1.

Let $\Gamma=\mathop{\rm Cay}\nolimits(G,S)$ be a distance-regular Cayley graph
with intersection array $\\{k,k-1,k-\mu;1,\mu,k\\}$, and $H$ be the part of
this bipartite graph which contains the identity element. Then $\Gamma$ can be
constructed on $H\rtimes\mathbb{Z}_{2}$, except possibly when $|H|=0\mod 4$,
$k$ is even, and $H$ is non-abelian.

Note that we are not claiming that $G$ is isomorphic to
$H\rtimes\mathbb{Z}_{2}$.

In the remainder of this section, we study the (possibly) exceptional cases
for the above theorem for certain well-known groups. Note that if $n=|H|$,
then $k(k-1)=(n-1)\mu$, so if $n$ is even, then $\mu$ must be even as well. So
we assume that $|H|=0\mod 4$, and $k$ and $\mu$ are even.

First, observe that the symmetric group has a unique normal subgroup of index
$2$, the alternating group, so also any bipartite Cayley graph on the
symmetric group, the part containing the identity element must be the
alternating group, and $G$ is isomorphic to $H\rtimes\mathbb{Z}_{2}$.

###  5.1. The dihedral group

Similarly as for the symmetric group, for $n$ odd, the dihedral group $D_{2n}$
has a unique normal subgroup of index $2$, the cyclic group (which is
trivially in accordance with the above proposition).

If $n$ is even, then $D_{2n}$ has three normal subgroups $H$ of index $2$ and
in each case, the group is the semidirect product of $H$ with $\mathbb{Z}_{2}$
[29, Thm. 3]. For the case where $H$ is not the cyclic group, we obtain
further information below. Note that this may be relevant for the
classification of distance-regular Cayley graphs on dihedral groups; see [27].

######  Proposition 5.2.

Let $\Gamma=\mathop{\rm Cay}\nolimits(G,S)$ be a distance-regular Cayley graph
with intersection array $\\{k,k-1,k-\mu;1,\mu,k\\}$, with $\mu<k-1$, such that
$G$ is a dihedral group of order $2n$, with $n=2m$ even and with cyclic
subgroup $C$ of order $n$. Let $H$ be the part of this bipartite graph which
contains the identity element. If $H$ is not the cyclic group, then the Cayley
graph $\Gamma_{C}=\mathop{\rm Cay}\nolimits(C,S\cap C)$ is the incidence graph
of a partial geometric design with parameters $(m,k_{1},\alpha,\beta)$ and
distinct eigenvalues $\\{\pm k_{1},\pm\sqrt{k-\mu},0\\}$, where $k_{1}=|S\cap
C|$, $k_{2}=k-k_{1}$, $\alpha=\frac{\mu(2k_{1}-k_{2})}{2}$, and
$2k_{1}-1+\beta-\alpha=(k_{1}-k_{2})^{2}=k-\mu$.

###### Proof.

The normal subgroup $C$ gives rise to an equitable partition with two parts
for this graph. Let $k_{1}=|S\cap C|$, $k_{2}=k-k_{1}$, then this partition
has quotient matrix

$\begin{bmatrix}k_{1}&k_{2}\\\ k_{2}&k_{1}\end{bmatrix}.$

Note that $k_{1}\neq 0$ because $C\neq H$ and $k_{2}\neq 0$ since $\Gamma$ is
connected. Furthermore, $(k_{1}-k_{2})^{2}=k-\mu$ since the eigenvalues
$k=k_{1}+k_{2}$ and $k_{1}-k_{2}$ of the quotient matrix are also eigenvalues
of the Cayley graph $\mathop{\rm Cay}\nolimits(G,S)$ [6, Lemma 2.3.1]. It then
also follows that $4k_{1}k_{2}=k(k-1)+\mu=n\mu$. Next, we will use that the
eigenvalues of Cayley graphs can be expressed as character sums of the
underlying group by a theorem of Babai [1, Thm. 3.1] (see also the survey
paper [23]), and use the relation between character sums of $G$ and those of
$C$.

The eigenvalues of the circulant Cayley graph $\Gamma_{C}$ are

$\lambda_{j}=\sum_{c\in S\cap C}\omega^{jc},$

where $j=0,1,\ldots,n-1$ and $\omega=e^{\frac{2\pi i}{n}}$ (for convenience,
we consider $S\cap C$ as a subset of $\mathbb{Z}_{n}$). Note that
$\lambda_{0}=k_{1}$, and in addition, it is easy to see that
$\lambda_{m+j}=-\lambda_{j}$ for every $j=0,1,\ldots,m-1$. On the other hand,
the irreducible characters of the dihedral group $G$ have degree $1$ or $2$.
The eigenvalues $\pm k$ (both have multiplicity $1$) of $\Gamma$ correspond to
irreducible characters of degree $1$. Thus, all other irreducible characters
must correspond to eigenvalues $\pm\sqrt{k-\mu}$ (and indeed, it is easy to
see that also $k_{1}-k_{2}$ corresponds to an irreducible character of degree
$1$).

Each of the $m-1$ irreducible characters $\psi_{j}$ of degree $2$ corresponds
to two eigenvalues $\lambda_{j1}$ and $\lambda_{j2}$ of $\Gamma$, and it
follows from Babai’s theorem, the character values, and the fact that $S$ is
inverse-closed that

$\lambda_{j1}+\lambda_{j2}=\sum_{s\in S}\psi_{j}(s)=\sum_{c\in S\cap
C}(\omega^{jc}+\omega^{-jc})=2\sum_{c\in S\cap C}\omega^{jc}=2\lambda_{j},$

for $j=1,\dots,m-1$.

Next, we use that $\lambda_{j1}$ and $\lambda_{j2}$ can only take values
$\pm\sqrt{k-\mu}$, so $\lambda_{j}$ can only take values $\pm\sqrt{k-\mu}$ and
$0$. This implies that the bipartite Cayley graph $\Gamma_{C}$ has (distinct)
eigenvalues $\\{\pm k_{1},0\\}$, $\\{\pm k_{1},\pm\sqrt{k-\mu}\\}$, or $\\{\pm
k_{1},\pm\sqrt{k-\mu},0\\}$.

If it has eigenvalues $\\{\pm k_{1},0\\}$, then it is a complete bipartite
graph. In this case $4mk_{2}=4k_{1}k_{2}=n\mu=2m\mu$ and therefore
$\mu=2k_{2}$. On the other hand, we have $k(k-1)=(n-1)\mu$. Hence
$(m+k_{2})(m+k_{2}-1)=(2m-1)2k_{2}$ and therefore $k_{2}=m-1$ or $k_{2}=m$,
which implies that $\Gamma$ is a complete bipartite graph minus a perfect
matching or a complete bipartite graph, which is a contradiction.

If $\Gamma_{C}$ has eigenvalues $\\{\pm k_{1},\pm\sqrt{k-\mu}\\}$, then it
must be distance-regular [6, Prop. 15.1.3]. On the other hand, distance-
regular circulant graphs have been classified [26] and therefore $\Gamma_{C}$
must be a complete bipartite graph minus a perfect matching. This implies
(from the eigenvalues) that $\mu=k-1$, and so $\Gamma$ is also a complete
bipartite graph minus a perfect matching, which again is a contradiction.

Thus, $\Gamma_{C}$ has eigenvalues $\\{\pm k_{1},\pm\sqrt{k-\mu},0\\}$ and so
it is the incidence graph of a partial geometric design with parameters
$(m,k_{1},\alpha,\beta)$, where $\alpha$ and $\beta$ are as stated (see [15,
§3.1] and Section 4), which completes the proof. ∎

As a side remark we note that that circulant distance-regular graphs have been
classified [26] (and the only non-trivial examples are Paley graphs), and
hence it follows that the induced graphs on $C$ cannot be distance-regular.

###  5.2. The dicyclic group

Let $G$ be the dicyclic group $Q_{4m}=\langle a,b\mid
a^{2m}=e,a^{m}=b^{2},b^{-1}ab=a^{-1}\rangle$, and let $n=2m$. If $m$ is odd,
then the group $G$ has a unique subgroup of index $2$, the cyclic group
$H=\langle a\rangle$ [29, Thm. 3]. In this case, we claim that there is no
non-trivial bipartite distance-regular Cayley graph with diameter $3$. Indeed,
suppose that $ba^{i}\in S$ for some $i$. Then also
$b^{-1}a^{i}=(ba^{i})^{-1}\in S$. But this implies that the two vertices $e$
and $b^{2}$ have the same neighbours, which is a contradiction. Therefore we
can conclude the following.

######  Proposition 5.3.

Let $m$ be odd. Then there is no non-trivial bipartite distance-regular Cayley
graph with diameter $3$ on the dicyclic groups $Q_{4m}$.

For $m$ even, the group $Q_{4m}$ has three subgroups of index $2$, and these
are cyclic or dicyclic [29, Thm. 3]. If $H$ is cyclic, then the same argument
as above implies that there is no non-trivial bipartite distance-regular
Cayley graph with diameter $3$ on the group $G$. If $H$ is dicyclic, then a
similar argument as in the proof of Proposition 5.2 applies. Therefore we can
conclude the following.

######  Proposition 5.4.

Let $\Gamma=\mathop{\rm Cay}\nolimits(G,S)$ be a distance-regular Cayley graph
with intersection array $\\{k,k-1,k-\mu;1,\mu,k\\}$, with $\mu<k-1$, such that
$G$ is the dicyclic group $Q_{4m}$ with cyclic subgroup $C$ of order $n=2m$.
Let $H$ be the part of this bipartite graph which contains the identity
element. If $H$ is not the cyclic group, then the Cayley graph
$\Gamma_{C}=\mathop{\rm Cay}\nolimits(C,S\cap C)$ is the incidence graph of a
partial geometric design with parameters $(m,k_{1},\alpha,\beta)$ and distinct
eigenvalues $\\{\pm k_{1},\pm\sqrt{k-\mu},0\\}$, where $k_{1}=|S\cap C|$,
$k_{2}=k-k_{1}$, $\alpha=\frac{\mu(2k_{1}-k_{2})}{2}$, and
$2k_{1}-1+\beta-\alpha=(k_{1}-k_{2})^{2}=k-\mu$.

We also note that if $m$ is even and $H$ is a dicyclic group, then the only
involution of the group $G=Q_{4m}$ is in $H$. So $S$ contains no involutions,
which implies that the Cayley graph $\mathop{\rm Cay}\nolimits(G,S)$ is not on
the semidirect product of a group — the part of this bipartite Cayley graph
which contains the identity element — and $\mathbb{Z}_{2}$.

###  5.3. The semidihedral group

Let $m=2^{\ell-1}$, with $\ell>1$. Let $G$ be the semidihedral group
$SD_{4m}=\langle a,b\mid a^{2m}=b^{2}=e,bab=a^{m-1}\rangle$. Then the group
$G$ has three subgroups of index $2$, the cyclic group, the dihedral group
$D_{2m}$, and the dicyclic group $Q_{2m}$ [29, Thm. 3]. If $H$ is cyclic or
dicyclic, then there exist involutions outside the normal subgroup $H$ and
therefore the group $G$ is the semidirect product of $H$ with
$\mathbb{Z}_{2}$.

The case that $H$ is a dihedral group is related to an old problem about the
(non)-existence of non-trivial difference sets in dihedral groups. It is
proved in [21] that if an $(n,k,\mu)$-difference set in a dihedral group of
order $n$ exists, then $k-\mu$ must be odd. Thus, Lemma 3.2 can be applied,
and we obtain the following.

######  Proposition 5.5.

Every bipartite distance-regular Cayley graph with diameter $3$ on a
semidihedral group is on the semidirect product of a group — the part of this
bipartite Cayley graph which contains the identity element — and
$\mathbb{Z}_{2}$.

We note that the character table of the semidihedral group is more complicated
than the character tables of the dihedral group and the dicyclic group, so to
extract a similar result as Propositions 5.2 and 5.4 seems quite ambitious.

###  5.4. Small cases

We finish this paper with some interesting small cases. The smallest
$2$-$(n,k,\mu)$ designs with $n$ and $k$ even and $k<n-1$ are the
$2$-$(16,6,2)$ designs, and their complementary $2$-$(16,10,6)$ designs. The
next smallest ones are the $2$-$(64,28,12)$ designs.

There are three non-isomorphic $2$-$(16,6,2)$ designs and therefore three non-
isomorphic distance-regular graphs with intersection array $\\{6,5,4;1,2,6\\}$
[5, p. 222]. It turns out that all three are Cayley graphs. One of them is the
folded $6$-cube which is a Cayley graph on the elementary abelian $2$-group of
order $32$ [13, §3.3]. By using GAP [11] and the difference sets in [19, Table
18.77], we can construct all three as follows.

  * •

$G_{1}$ is the elementary abelian group of order $32$ with generators $\langle
a,b,c,d,f\rangle$ of involutions, $H=\langle a,b,c,d\rangle$, and
$S=\\{af,bf,cf,df,f,abcdf\\}$;

  * •

$G_{2}=\langle a,b,c\mid
a^{8}=b^{2}=c^{2}=e,ab=ba,cac=a^{-1},cbc=b^{-1}\rangle$ is isomorphic to
$\mathop{\rm Dih}\nolimits(H)$, $H=\langle a,b\rangle$ is isomorphic to
$\mathbb{Z}_{8}\times\mathbb{Z}_{2}$, and
$S=\\{c,ca,ca^{2},ca^{4},cab,ca^{6}b\\}$;

  * •

$G_{3}=\langle a,b,c,d\mid
a^{4}=c^{2}=d^{2}=(da)^{2}=(da^{2})^{2}=e,a^{2}=b^{2},ac=ca,bc=cb,dabc=(db)^{-1}\rangle$
is isomorphic to $SD_{16}\times\mathbb{Z}_{2}$, $H=\langle a,b,c\rangle$ is
isomorphic to $Q_{8}\times\mathbb{Z}_{2}$ and
$S=\\{d,da^{2},(db)^{4}da,\break(db)^{6}da,db,dabc\\}.$

This implies that every bipartite distance-regular graph with diameter $3$ on
$32$ vertices is a Cayley graph on the semidirect product of a group — the
part of this bipartite Cayley graph which contains the identity element — and
$\mathbb{Z}_{2}$. It turns out that all three distance-regular graphs have the
$4$-cube as an induced subgraph. This antipodal bipartite distance-regular
graph with diameter $4$ can (once more) be constructed as a Cayley graph
$\mathop{\rm Cay}\nolimits(K,S\cap K)$ on a subgroup $K$ of index $2$ — cf.
Propositions 5.2 and 5.4 — as described below (see also the structures of
connected $4$-regular bipartite integral Cayley graphs in [28, Table 3]).
Recall also Proposition 4.2.

  * •

$K=\langle af,bf,cf,df\rangle$ is an elementary abelian subgroup in $G_{1}$,
and $S\cap K=\\{af,bf,cf,df\\}$;

  * •

$K=\langle a^{2},b,c\rangle\cong D_{8}\times\mathbb{Z}_{2}$ is a subgroup in
$G_{2}$, and $S\cap K=\\{c,ca^{2},ca^{4},ca^{6}b\\}$;

  * •

$K=\langle db,da\rangle\cong SD_{16}$ is a subgroup in $G_{3}$, and $S\cap
K=\\{(db)^{4}da,(db)^{6}da,db,dabc\\}$.

## Acknowledgements

The research of Mojtaba Jazaeri was in part supported by a grant from School
of Mathematics, Institute for Research in Fundamental Sciences (IPM) (No.
96050014). The authors thank Robert Bailey for [3].

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2022

[1,2]Marcus Scheck

[1]School of Computing, Engineering, and Physical Sciences, University of the
West of Scotland, High Street, Paisley, PA11 3PX, UK

2]SUPA, Scottish Universities Physics Alliance, UK

3]Department of Physics, University of York, Heslington, YO10 5DD, UK

4]Department of Physics, University of Warsaw, Warsaw, PL-02-093, Poland

5]Materials Theory, ETH Zürich, Zürich, CH-8093, Switzerland

6] National Physical Laboratory, Teddington, TW11 0LW, UK

7]Biological and Environmental Sciences, University of Stirling, Stirling, FK9
4LA, UK

# A new avenue in the search for CP violation: Mössbauer spectroscopy of 227Ac

<EMAIL_ADDRESS>Robert Chapman  Jacek Dobaczewski  Claude Ederer
Peter Ivanov  Guiseppe Lorusso  David O’Donnell  Christian Schröder  *  [  [
[  [  [  [

###### Abstract

This work proposes a new avenue in the search for CP-violating odd-electric
and even-magnetic nuclear moments. A promising candidate to find such moments
in the ground state is the quadrupole-deformed and octupole-correlated nucleus
227-actinium. In this nucleus, the 27.4-keV $E1$ transition that connects the
$3/2^{+}$ parity-doublet partner and the $3/2^{-}$ ground state is perfectly
suited to apply the sensitive technique of recoil-free selfabsorption,
commonly known as Mössbauer spectroscopy. In this experimental approach, the
lifetime of the $3/2^{+}$ upper parity-doublet partner allows an estimate of
the lower limit of $\Delta E=2\cdot\Gamma_{\gamma}$= $23.7(1)\times 10^{-9}$
eV for the achievable energy resolution to be made. This resolution must be
exceeded by the interaction of a CP-violating moment and the corresponding
multipole moment of the field distribution in the lattice. This work presents
the first ideas for patterns caused by CP-violating moments on the expected
quadrupole splitting and nuclear Zeeman effect.

###### keywords:

Quadrupole-octupole coupling, CP-violating moments, Mössbauer spectroscopy,
Quadrupole spliting, Nuclear Zeeman effect

##  1 Introduction

The observation of enhanced $B(E3,0^{+}\rightarrow 3^{-}_{1})$ excitation
strength in several lanthanide Bucher1 ; Bucher2  and actinide Spagnoletti ;
Butler ; Gaffney ; Wolle  isotopes indicates octupole correlations in the
quadrupole-deformed ground state of at least some of these nuclei. The
resulting quadrupole-octupole deformed pear shape is predicted to enhance the
possible CP-violating laboratory Schiff moment Spevak ; Auerbach ; Naftali ;
Zelevinsky ; Dobacewski ; Markus ; ViktorM2 .

Interestingly, a long-standing theme investigated in
$(\gamma,\gamma^{\prime})$ photon-scattering experiments is the $E1$ strength
of the $[2^{+}\otimes 3^{-}]_{1^{-}}$ quadrupole-octupole coupled (QOC)
$1^{-}_{1}$ levels Kneissl96 . In particular, Kneissl, Pitz, and coworkers
established in stable nuclei comprehensive systematics of these, in general,
lowest-lying 1${}^{-}_{1}$ levels Fransen ; Kneissl07 . The collective nature
of these $1^{-}_{1}$ levels is evidenced by their energy systematics (Fig. 1);
these systematics display a smooth behaviour, which can be summarised as
follows. At/near closed shells, the energy $E_{1^{-}}$ of the QOC 1- level
corresponds nearly to the sum $\Sigma=E_{2^{+}_{1}}+E_{3^{-}_{1}}$ of the
excitation energies of the first $2^{+}_{1}$ and $3^{-}_{1}$ levels, but
decreases relative to $\Sigma$ with the onset of quadrupole correlations.
Transitional nuclei with ground-state quadrupole correlations, but no well
developed quadrupole deformation, exhibit a near degeneracy of $1^{-}_{1}$ and
$3^{-}_{1}$ levels. Once static quadrupole deformation is present, levels
1${}^{-}_{1}$ and $1^{-}_{2}$ become the band-heads of the $K=0$ and $K=1$
octupole bands. Furthermore, the $B(E1,0^{+}\rightarrow 1^{-}_{1})$ strength
remains in the same order of magnitude over a wide range of nuclei with
varying underlying quadrupole deformation (Fig. 2). In addition, in spherical
nuclei it has been shown that the strength of the two-phonon
creating/annihilating transition connecting this $1^{-}$ state with the ground
state scales with the $3^{-}_{1}\rightarrow 2^{+}_{1}$ two-phonon exchanging
transition Pietralla97 , whereas in well-deformed nuclei the branching
behaviour predicted by the Alaga rules is observed ZilgesALAGA .
Interestingly, an enhanced E1 strength of up to $20\times 10^{-3}$e2fm2 is
noticeable for semi-magic nuclei and prolate-deformed nuclei, which is almost
an order of magnitude reduced in the transitional region.

Figure 1:  Systematics of the excitation energy $E_{1^{-}}$ of the first
excited $1^{-}_{1}$ level in the lanthanide/rare-earth region. Figure is taken
from Ref. ScheckHf .

Figure 2:  Systematics of $B(E1,0^{+}_{gs}\rightarrow 1^{-}_{1})$ strength in
the lanthanide/rare-earth region. Figure is taken from Ref. ScheckHf .

In spherical nuclei, for which the $2^{+}_{1}$ and $3^{-}_{1}$ levels are
interpreted as phonons, the QOC $1^{-}$ level is the low-spin member of a
quintuplet of negative-parity levels with spins ranging from $J^{\pi}=1^{-}$
to $5^{-}$. However, candidate levels for the full multiplet are proposed for
only a few nuclei, see, e.g., Refs. Wilhelm ; JohnPaul . While quadrupole
deformation in the ground state of the nuclear many-body quantum system is
well established, only a few candidates with enhanced octupole correlations
and possibly even deformation were proposed following the observation of
enhanced $B(E3,0^{+}\rightarrow 3^{-}_{1})$ strength in Coulomb-excitation
experiments Butler ; Gaffney ; Wolle .

Figure 3 shows the inverse energy-weighted $B(E3)$ strengths. This quantity
combines the two most relevant characteristics of octupole
correlations/deformations and emphasises the special nature of the radium
nuclei 222,224,226Ra. Furthermore, for 228Th, indirect experimental evidence
Majid  suggests enhanced octupole correlations. The interplay of quadrupole
deformation and octupole correlations, for the above mentioned nuclei in the
ground state, results in the nucleus adopting a pear shape, in which the odd-
electric (E1, E3, and E5) and even-magnetic (M2 and M4) moments are present in
the intrinsic reference frame. Indeed, quadrupole-octupole coupling is
predicted to enhance a possible nuclear Schiff moment Spevak ; Auerbach ;
Naftali ; Zelevinsky ; Dobacewski ; Markus  and magnetic quadrupole ($M2$)
moment ViktorM2  caused by CP-violating physics Musolf ; Tim .

Figure 3:  Inverse energy-weighted $B(E3,0^{+}\rightarrow 3^{-}_{1})$ strength
as a function of the mass number $A$. The systematics include nuclei for which
the $B(E3)$ strength is known Kibedi . The recent data points for 144Ba
Bucher1 , 146Ba Bucher2 , 220,222Rn Gaffney ; Spagnoletti , and
222,224,226,228Ra Butler ; Gaffney ; Wolle  are highlighted.

The projection of the pear shape PeterWitek  from the intrinsic to the
laboratory reference frame results in good-parity wave functions, that are
linear combinations of the pear shape pointing to the left $\Psi_{l}$ and to
the right $\Psi_{r}$. Assuming that $\Psi_{l}$ and $\Psi_{r}$ are orthogonal,
the linear combinations are

$\Psi^{+}=\frac{1}{\sqrt{2}}\left(\Psi_{l}+\Psi_{r}\right),\quad\quad\Psi^{-}=\frac{1}{\sqrt{2}}\left(\Psi_{l}-\Psi_{r}\right).$
(1)

The linear combination $\Psi^{+}$ is invariant under the space-inversion $P$
operation. However, the $\Psi^{-}$ linear combination inverts its sign and,
therefore, has negative parity $\pi=-$. In an odd-mass nucleus, the coupling
of the unpaired particle to these two states leads to the presence of parity
doublets, which are two levels with identical angular momentum but opposite
parity. Indeed, as shown in Table 1, several odd-mass nuclei in the region
near $Z=88/90$ and $N=134/136$ exhibit parity-doublet candidates. Clearly, the
data in Table 1 suffer from uncertainties concerning spin and, especially,
parity assignment, and unobserved upper partner levels. Nevertheless, at
present 227Ac is the nucleus with the lowest established parity-doublet energy
difference $\Delta E_{PD}$ of 27.4 keV.

In phenomenological estimates (see Refs. Musolf ; Tim  and references
therein), the enhancement of the intrinsic nuclear Schiff moment,
$\langle\hat{S}\rangle$, is predicted to scale with the quadrupole $\beta_{2}$
and the square of the octupole $\beta_{3}$ deformations, as well as inversely
with the energy difference $\Delta E_{PD}$ between the parity doublet partners

$\langle\hat{S}\rangle\propto\frac{\beta_{2}(\beta_{3})^{2}}{\Delta E_{PD}}.$
(2)

The quadrupole-octupole coupling contributes with the factor
$\beta_{2}\cdot\beta_{3}$ and the expected CP-violating interaction with an
additional $\beta_{3}$ factor. However, at present it is not possible to
disentangle the static and dynamic contributions to the $\beta_{3}$
deformation parameter in a model-independent way. Nevertheless, self-
consistent calculations in well quadrupole-octupole deformed nuclei allow a
determination of $\langle\hat{S}\rangle$ values directly, without passing
through the estimates of deformations $\beta_{2}$ and $\beta_{3}$.

Table 1:  Data for selected odd-mass nuclei in the $A\approx 224$ mass region
exhibiting parity-doublet candidates. The table presents the isotope, its
halflife $T_{1/2,gs}$, spin and parity of the ground state $J_{0}^{\pi}$,
energy difference to the lowest-lying possible parity-doublet partner $\Delta
E_{PD}$, and the lifetime $T_{1/2,ul}$ of the upper level. Data are taken from
the NNDC data base NNDC  and Ref. PTB .

Nucleus | $T_{1/2,gs}$ | $J^{\pi}$ | $\Delta E_{PD}$ | $T_{1/2,ul}$
---|---|---|---|---
|  |  | [keV] | [ns]
223Fr | 22.00(7) m | 3/2(-) | 134.48(4) |
225Fr | 3.95(14) m | 3/2- | 142.59(3) |
227Fr | 2.47(3) m | 1/2+ | 59.10(5) |
221Ra | 28(2) s | 5/2+ | 103.61(11) |
223Ra | 11.43(5) d | 3/2+ | 50.128(9) | 0.63(7)
225Ra | 14.9(2) d | 1/2+ | 55.16(6) |
227Ra | 42.2(5) m | 3/2+ | 90.034(2) | 0.254(9)
223Ac | 2.10(5) m | (5/2-) | 64.62(4) |  $\leq$ 0.250
225Ac | 9.920(3) d | (3/2-) | 40.10(4) | 0.72(3)
227Ac | 21.772(3) y | 3/2- | 27.369(11) | 38.52(19)a
229Ac | 62.7(5) m | (3/2+) | 104.3(4) |
229Th | 7880(120) y | 5/2+ | 146.357(2) |
231Th | 25.52(1) h | 5/2+ | 185.718(2) | 1.07(8)
229Pa | 1.50(5) d | (5/2+) | 99.3(4) |
231Pa | 32760(11) y | 3/2- | 102.269(2) |  $\leq$ 0.7

aThis value is taken from the most recent evaluation PTB .

The above mentioned enhanced $B(E3)$ probabilities may indicate the presence
of a non-zero intrinsic $E3$ moment in the ground state of Ra isotopes and,
consequently, a possibility of the additional non-zero intrinsic $E1$ and $E5$
as well as $M2$ and $M4$ moments. We can thus expect that a CP-violating
interaction may induce enhanced values of the corresponding symmetry-violating
laboratory moments.

##  2 Experimental motivation for studying 227Ac

Given the high charge number of $Z=89$ and low energy of the transition
connecting the parity-partner levels in 227Ac, and considering that the
conversion coefficients (CCs) exhibit a strong multipolarity dependence, a
measurement of the CCs appear to be the obvious way to search for CP-violating
physics. If parity was no longer a good quantum number, the $E1$ transition
connecting the parity-doublet partners would contain a $M1$ component.
However, it can be assumed that the P-/T-odd effect scales as a $10^{-7}$
contribution of the weak interaction to the nuclear force. Hence, any signal
of such physics will be well below the associated uncertainty in the
calculation of CCs. For the 27.4-keV $E1$ transition in 227Ac, the value of
the CC is $\alpha_{C}=3.54(5)$ BRICC , which still allows for $\approx 22$ %
of all decays to proceed via $\gamma$-ray emission.

The low $\gamma$-ray energy, and hence the low momentum transfer in the
emission and absorption process, enables the nuclear photonics technique of
recoil-free resonant absorption (known as Mössbauer spectroscopy Moessbauer ;
Guetlich1 ; Guetlich2 ) to be used as a method to investigate the $E1$
transition in question and, subsequently, to pin down the properties of the
two parity-doublet partners. Mössbauer spectroscopy exploits the fact that,
for a nucleus embedded in a crystal lattice, there is a probability that the
recoil momentum transfer in the emission and absorption of a $\gamma$ ray is
absorbed by the entire crystal. If both processes are recoil-free, the
theoretical achievable energy-resolution $\Delta E\approx
2\cdot\Gamma_{\gamma}$ is limited only by the natural line width $\Gamma$,
which for 227Ac is
$\Gamma_{\gamma}=\frac{\hbar\ln(2)}{T_{1/2,ul}}=11.8(1)\times 10^{-9}$ eV.

Other advantages of studying 227Ac are its presence in the decay chain of 235U
and a comparably long half-life ($T_{1/2}\approx 21.8$ years NNDC ). These may
allow a chemical separation of a sufficient amount of the target material to
manufacture an absorber target, whose lattice would be tailored to the
specific requirements of Mössbauer spectroscopy.

Mössbauer spectroscopy can be performed in fluorescence or absorption. The
latter, for which the detector is situated in the extension of the emitter-
absorber line, has the advantage that the detector can be retracted from the
absorber/emitter samples. This prevents pile-up events due to a too high
detector count rate. Given the relatively small expected size of the
radioactive samples, the resulting reduction of the solid angle would not
adversely impact the count rate of good events.

A further benefit of 227Ac is that the upper parity-doublet level is strongly
populated following either the $\alpha$ decay of 231Pa ($T_{1/2}=32760(110)$
years) Teoh  or the 227Ra $\beta$ decay ($T_{1/2}=42.2(5)$ min) Lourens .
While practical considerations involving the lifetime of 231Pa favour the
population via $\alpha$ decay, the recoil experienced by the 227Ac daughter
might be devastating in terms of a well-defined position of the emitting 227Ac
nucleus within the lattice structure and, therefore, local field distribution.

##  3 CP-violating moments and Mössbauer spectroscopy

Traditionally, three effects can be observed in Mössbauer spectroscopy,
namely, isomer shift, quadrupole splitting, and nuclear Zeeman effect. Each of
them slightly shifts the energies of the involved levels and, consequently,
alters the energy of the emitted and/or absorbed $\gamma$ rays. The resulting
energy difference can be compensated by mounting the emitter or absorber
source on a drive inducing a velocity-dependent Doppler shift. For example,
for 227Ac an energy shift of $11.74\times 10^{-9}$ eV corresponds to a sample
velocity of $1$ cm/s.

Figure 4:  Part a) shows the level scheme of a nucleus with a
$J^{\pi}=3/2^{\pm}$ parity doublet under the influence of quadrupole
splitting. Part b) includes the influence of the interaction $V_{oE}$ of an
$EL$ odd-electric moment with the corresponding $L^{th}$ derivative of the
electric potential of the crystal lattice at the nuclear coordinates. This CP-
violating physics lifts the $|m|$ degeneracy of the time-reversal orbits with
magnetic quantum numbers $m$ and $-m$. For a detailed discussion see text.

The isomer shift corresponds to a slight shift of the Coulomb energy of a
nuclear level due to the electron density and, consequently, electrostatic
potential at the nuclear coordinates; it is observed, if emitter and absorber
nuclei are embedded in different crystal lattices. The second effect of
quadrupole splitting is observed if, due to the crystal composition at the
coordinates of the absorbing nucleus, an electric-field gradient is present.
This situation is shown in Fig. 4 a) for a nucleus with a $J^{\pi}=3/2^{\pm}$
parity doublet. The reflection symmetry of the quadrupole shape results in a
$|m|$ degeneracy of the time-reversal $m$ and $-m$ orbits. For identical
quadrupole moments of the parity-doublet partners, and, therefore, identical
splitting, the spectrum would contain three transitions, with $\Delta|m|=\pm
1$ shifted to lower/higher energy and the $\Delta|m|=0$ transitions at the
unperturbed transition energy. However, due to the parity-doublet partners
being linear combinations, it can be expected that their quadrupole moments
differ and the transition energies of the two $\Delta|m|=0$ lines will be
different. Hence, Mössbauer spectroscopy represents a sensitive test to
extract at least the relative difference of the quadrupole moments or, since
the quadrupole moment of the $J^{\pi}=3/2^{-}$ ground state is known, the
quadrupole moment of the upper partner level. For such a test it is of benefit
that line-shape parameters, especially the Full-Width at Half Maximum, can be
fitted to the two $\Delta|m|=\pm 1$ transitions and even a small widening of
the central line consisting of the two $\Delta|m|=0$ transitions would allow
an extraction of the difference.

Far more intriguing is that, if the investigated nucleus is truly reflection
asymmetric, any residual interaction of an odd-electric moment with a higher
moment of the multipole expansion of the lattice charge distribution at the
nuclear coordinates (e.g., $E1$ moment with the electric field or the $E3$
moment with the curvature of the electric field) would lift the $|m|$
degeneracy. Again, given that $EL$ moments of the two physical states
represented by the linear combinations differ, a different interaction
$V_{oE}$ of $EL$ odd-electric moment and $L^{th}$ multipole order of the
lattice charge distribution can be anticipated for each doublet partner level.
Consequently, a split in the transitions will be observed. With the examption
of the $\Delta m=\pm 1$ transitions of the $\Delta|m|=0$, namely the
$m_{\text{upper}}=\pm 1/2\rightarrow m_{\text{lower}}=\mp 1/2$ transitions,
for all transitions the splitting results in the same energy shift. Only these
$\Delta m=\pm 1$ transitions experience a stronger shift and the $\Delta|m|=0$
transition will, in addition to the quadrupole splitting exhibit a further
splitting into a quadruplet. In the quadruplet the two satellite lines are
expected to show half the intensity of the lesser shifted $\Delta|m|=0$,
$\Delta m=0$ transitions. It is helpful that the two $\Delta|m|=\pm 1$ lines
are not affected in the same way and allow a fit of the peak shape.
Consequently, even if $V_{oE}$ is less than $\Delta E$, an alteration of the
central line’s peak shape yields evidence for CP-violating physics possibly
even an order of magnitude below the experimental resolution $\Delta E$.
Concerning the interaction of a nuclear $E1$ moment and the electric field at
the nuclear coordinates, it must be mentioned that, in a given lattice
structure, the nucleus will position itself at coordinates for which the net
electric field vanishes. Therefore, the $E3$ moment should be the lowest odd-
electric moment contributing. Eventually, the use of a piezzo-electric lattice
will allow access to the $E1$ moment.

Finally, the combination of the sensitivity of Mössbauer spectroscopy and the
magnetic field at the nuclear coordinates in a lattice allow the observation
of the nuclear Zeeman effect. If the magnetic moments of the parity-doublet
partners are identical, the three-line pattern of the normal Zeeman effect
will be observed. However, due to the two parity partners being linear
combinations, their magnetic properties differ and the pattern of the
anomalous Zeeman effect is expected. Here, for a $J=3/2$ parity doublet, ten
transitions can be observed. However, these lines will exhibit a centroid
symmetry. Given it is present, the interaction of a CP-forbidden magnetic
quadrupole $M2$ moment ViktorM2  with the magnetic-field gradient perturbs the
expected pattern. Since the additional interaction is direction-dependent, it
can be expected that the otherwise $m$-independent Zeeman splitting between
two levels with $m=0$ and $m\pm 1$ receives an additional $m$-dependent term.
For such a $m$-dependent splitting, the centroid-symmetric pattern will be
disturbed.

To extract a quantitative result for a possible CP-violating interaction,
knowledge of the electric and magnetic field distribution, specifically the
field values and higher derivatives at the position of the nucleus, is
required. These values can be calculated for specific materials using modern
density functional theory Martin , and this has already been successfully used
to obtain electric field gradient and hyperfine fields for a variety of
materials (see, e.g., Bluegel ; Blaha1 ; Blaha2 ).

##  4 Summary and Outlook

To summarise, in this contribution, we proposed Mössbauer spectroscopy of the
octupole-correlated 227Ac nucleus as a new avenue to search for CP-violating
physics. This work provides first thoughts in relation to the way in which a
residual interaction associated with an odd-electric or even-magnetic moment
alters the effects of quadrupole splitting or the nuclear Zeeman effect. In a
long-term future outlook, this project would benefit from a sufficiently mono-
energetic source with $\Delta E_{\gamma}\ll\Gamma$ of a fully-polarised
$\gamma$-ray beam. This would allow the elimination of effects associated with
recoils in the radioactive decays and in addition the selective population of
$m$ substates can be used to test the involved $m$ substates.

Acknowledgments

We acknowledge financial support by the UK-STFC Grant Nos. ST/P005101/1,
ST/P003885/1, and ST/V001035/1, by the Polish National Science Centre under
Contract No. 2018/31/B/ST2/02220, and by a Leverhulme Trust Research Project
Grant. We acknowledge the CSC-IT Center for Science Ltd., Finland, for the
allocation of computational resources. This project was partly undertaken on
the Viking Cluster, which is a high performance computing facility provided by
the University of York. We are grateful for computational support from the
University of York High Performance Computing service, Viking and the Research
Computing team.

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