# HS-integral and Eisenstein integral normal mixed Cayley graphs

Monu Kadyan and Bikash Bhattacharjya
Department of Mathematics
Indian Institute of Technology Guwahati, India
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Abstract

A mixed graph is called _second kind hermitian integral_(or _HS-integral_) if
the eigenvalues of its Hermitian-adjacency matrix of second kind are integers.
A mixed graph is called _Eisenstein integral_ if the eigenvalues of its (0,
1)-adjacency matrix are Eisenstein integers. A mixed Cayley graph
$Cay(\Gamma,S)$ is called normal if $S$ is the union of some conjugacy classes
of the group $\Gamma$. We characterize the set $S$ for which a normal mixed
Cayley graph $\text{Cay}(\Gamma,S)$ is HS-integral. We also show that that
every Eisenstein integral normal mixed Cayley graph is an HS-integral. We
purpose a conjecture to prove its converse. In short, this paper extends
almost all the results of Kadyan-Bhattarcharjya to normal mixed Cayley graphs.

Keywords. integral graphs; HS-integral mixed graph; Eisenstein integral mixed
graph; normal mixed Cayley graph.
Mathematics Subject Classifications: 05C50, 05C25

##  1 Introduction

A _mixed graph_ $G$ is a pair $(V(G),E(G))$, where $V(G)$ and $E(G)$ are the
vertex set and the edge set of $G$, respectively. Here $E(G)\subseteq
V(G)\times V(G)\setminus\\{(u,u)~{}|~{}u\in V(G)\\}$. If $G$ is a mixed graph,
then $(u,v)\in E(G)$ need not imply that $(v,u)\in E(G)$. An edge $(u,v)$ of a
mixed graph $G$ is called undirected if both $(u,v)$ and $(v,u)$ belong to
$E(G)$. An edge $(u,v)$ of a mixed graph $G$ is called directed if $(u,v)\in
E(G)$ but $(v,u)\notin E(G)$. A mixed graph can have both undirected and
directed edges. A mixed graph $G$ is said to be a simple graph if all the
edges of $G$ are undirected. A mixed graph $G$ is said to be an oriented graph
if all the edges of $G$ are directed.

For a mixed graph $G$ on $n$ vertices, its (0, 1)-adjacency matrix and
Hermitian-adjacency matrix of the second kind are denoted by
$\mathcal{A}(G)=(a_{uv})_{n\times n}$ and $\mathcal{H}(G)=(h_{uv})_{n\times
n}$, respectively, where

$a_{uv}=\left\\{\begin{array}[]{rl}1&\mbox{ if }(u,v)\in E\\\ 0&\textnormal{
otherwise,}\end{array}\right.~{}~{}~{}~{}~{}\text{ and
}~{}~{}~{}~{}~{}~{}h_{uv}=\left\\{\begin{array}[]{cl}1&\mbox{ if }(u,v)\in
E\textnormal{ and }(v,u)\in E\\\ \frac{1+i\sqrt{3}}{2}&\mbox{ if }(u,v)\in
E\textnormal{ and }(v,u)\not\in E\\\ \frac{1-i\sqrt{3}}{2}&\mbox{ if
}(u,v)\not\in E\textnormal{ and }(v,u)\in E\\\ 0&\textnormal{
otherwise.}\end{array}\right.$

The Hermitian-adjacency matrix of the second kind was introduced by Bojan
Mohar [26].

Let $G$ be a mixed graph. By an _HS-eigenvalue_ of $G$, we mean an eigenvalue
of $\mathcal{H}(G)$. By an _eigenvalue_ of $G$, we mean an eigenvalue of
$\mathcal{A}(G)$. Similarly, the _HS-spectrum_ of $G$, denoted $Sp_{H}(G)$, is
the multi-set of the HS-eigenvalues of $G$, and the _spectrum_ of $G$, denoted
$Sp_{A}(G)$, is the multi-set of the eigenvalues of $G$. Note that the
Hermitian-adjacency matrix of the second kind of a mixed graph is a Hermitian
matrix, and so its HS-eigenvalues are real numbers. However, if a mixed graph
$G$ contains at least one directed edge, then $\mathcal{A}(G)$ is non-
symmetric. Accordingly, the eigenvalues of $G$ need not be real numbers. The
matrix obtained by replacing $\frac{1+i\sqrt{3}}{2}$ and
$\frac{1-i\sqrt{3}}{2}$ by $i$ and $-i$, respectively, in $\mathcal{H}(G)$, is
called the _Hermitian adjacency_ matrix of $G$. Hermitian adjacency matrix of
mixed graphs was introduced in [13, 24].

A mixed graph is called H-integral if the eigenvalues of its Hermitian
adjacency matrix are integers. A mixed graph $G$ is said to be HS-integral if
all the HS-eigenvalues of $G$ are integers. A mixed graph $G$ is said to be
Eisenstein integral if all the eigenvalues of $G$ are Eisenstein integers.
Note that complex numbers of the form $a+b\omega_{3}$, where
$a,b\in\mathbb{Z},\omega_{3}=\frac{-1+i\sqrt{3}}{2}$, are called _Eisenstein_
integers. An HS-integral simple graph is called an _integral_ graph. Note that
$\mathcal{A}(G)=\mathcal{H}(G)$ for a simple graph $G$. Therefore in case of a
simple graph $G$, the terms HS-eigenvalue, HS-spectrum and HS-integrality of
$G$ are the same with that of eigenvalue, spectrum and integrality of $G$,
respectively.

Integrality of simple graphs have been extensively studied in the past.
Integral graphs were first defined by Harary and Schwenk [14] in 1974 and
proposed a classification of integral graphs. See [4] for a survey on integral
graphs. Watanabe and Schwenk [32, 33] proved several interesting results on
integral trees in 1979. Csikvari [10] constructed integral trees with
arbitrary large diameters in 2010. Further research on integral trees can be
found in [7, 6, 30, 31]. In $2009$, Ahmadi et al. [2] proved that only a
fraction of $2^{-\Omega(n)}$ of the graphs on $n$ vertices have an integral
spectrum. Bussemaker et al. [8] proved that there are exactly $13$ connected
cubic integral graphs. Stevanović [29] studied the $4$-regular integral graphs
avoiding $\pm 3$ in the spectrum, and Lepović et al. [22] proved that there
are $93$ non-regular, bipartite integral graphs with maximum degree four. In
2017, Guo et. al. [13] found all possible mixed graphs on $n$ vertices with
spectrum $\\{-n+1,-1,-1,...,-1\\}$.

Throughout the paper, we consider $\Gamma$ to be a finite group. Let $S$ be a
subset, not containing the identity element, of $\Gamma$. The set $S$ is said
to be symmetric (resp. skew-symmetric) if $S$ is closed under inverse (resp.
$a^{-1}\not\in S$ for all $a\in S$). Define $\overline{S}=\\{u\in
S:u^{-1}\not\in S\\}$. Clearly, $S\setminus\overline{S}$ is symmetric and
$\overline{S}$ is skew-symmetric. The mixed Cayley graph $G=Cay(\Gamma,S)$ is
a mixed graph, where $V(G)=\Gamma$ and $E(G)=\\{(a,b):a,b\in\Gamma,ba^{-1}\in
S\\}$. If $S$ is symmetric then $G$ is a simple Cayley graph. If $S$ is skew-
symmetric then $G$ is an oriented Cayley graph. A mixed Cayley graph
$Cay(\Gamma,S)$ is called normal if $S$ is the union of some conjugacy classes
of the group $\Gamma$.

If every element of a group is conjugate to its inverse, the group is said to
be ambivalent. All symmetric groups $S_{n}$ are ambivalent, as is widely
known. The alternating groups $A_{n}$ are ambivalent only for
$n\in\\{1,2,5,6,10,14\\}$. It is clear that all normal mixed Cayley graphs
over an ambivalent group are undirected. Thus, in order to examine the
integrality of normal mixed Cayley graph with at least one directed edge, we
must look at non-ambivalent groups. A basic example of a non-ambivalent group
is an abelian group.

In 1982, Bridge and Mena [5] introduced a characterization of integral Cayley
graphs over abelian groups. Later on, same characterization was rediscovered
by Wasin So [27] for cyclic groups in 2005. In 2009, Abdollahi and Vatandoost
[1] proved that there are exactly seven connected cubic integral Cayley
graphs. On the same year, Klotz and Sander [20] proved that if Cayley graph
$Cay(\Gamma,S)$ over an abelian group $\Gamma$ is integral then $S$ belongs to
the Boolean algebra $\mathbb{B}(\Gamma)$ generated by the subgroups of
$\Gamma$. Moreover, they conjectured that the converse is also true, which was
proved by Alperin and Peterson [3]. In 2014, Cheng et al. [21] proved that
normal Cayley graphs over symmetric groups are integral. In 2017, Lu et al.
[25] gave necessary and sufficient condition for the integrality of Cayley
graphs over the dihedral group $D_{n}=\langle
a,b|a^{n}=b^{2}=1,bab=a^{-1}\rangle$. In particular, they completely
determined all integral Cayley graphs over the dihedral group $D_{p}$ for a
prime $p$. In 2019, Cheng et al.[9] obtained several simple sufficient
conditions for the integrality of Cayley graphs over the dicyclic group
$T_{4n}=\langle a,b|a^{2n}=1,a^{n}=b^{2},b^{-1}ab=a^{-1}\rangle$. In
particular, they also completely determined all integral Cayley graphs over
the dicyclic group $T_{4p}$ for a prime $p$.

H-integrality of mixed Cayley graphs over abelian groups have been discussed
in [18] and [19]. HS-integrability and Eisenstein integrability of mixed
Cayley graphs over abelian groups have been discussed in [16] and [17]. In
2014, Godsil _et al._ [12] characterised integral normal Cayley graphs. In
this paper, we characterise the HS-integrality and Eisenstein integrality of
normal mixed Cayley graphs in terms of their connection set.

The following is how this document is structured. Some preliminary notions and
results are addressed in Section 2. We express the HS-eigenvalues of a normal
mixed Cayley graph as the sum of the HS-eigenvalues of a simple Cayley graph
and an oriented Cayley graph. In Section 3, we characterise the HS-integrality
of normal oriented Cayley graphs in terms of symbol set. In Section 4, we
extend this characterisation to normal mixed Cayley graphs. In the last
section, we discuss the Eisenstein integrality of normal mixed Cayley graphs.

##  2 Preliminaries

Let $\Gamma$ be a finite group and $g,h\in\Gamma$. If $g=x^{-1}hx$ for some
$x\in\Gamma$ then we say that $h$ is a conjugate of $g$. The set of all
conjugates of $g$ in $\Gamma$ is called the conjugacy class of $g$ and is
denoted by $Cl(g)$. Define $C_{\Gamma}(g)$ to be the set of all elements of
$\Gamma$ that commute with $g$. We denote the group algebra of $\Gamma$ over
the field $\mathbb{Q}(\omega_{3})$ by $\mathbb{Q}(\omega_{3})\Gamma$. That is,
$\mathbb{Q}(\omega_{3})\Gamma$ is the set of all sums
$\sum\limits_{g\in\Gamma}c_{g}g$, where $c_{g}\in\mathbb{Q}(\omega_{3})$ and
we assume $1.g=g$ to have $\Gamma\subseteq\mathbb{Q}(\omega_{3})\Gamma$.

A representation of a finite group $\Gamma$ is a homomorphism $\rho:\Gamma\to
GL(V)$ for some finite-dimensional vector space $V$ over the complex field
$\mathbb{C}$, where $GL(V)$ denotes the group of automorphisms of $V$. The
dimension of $V$ is called the degree of $\rho$. Two representations
$\rho_{1}$ and $\rho_{2}$ of $\Gamma$ on $V_{1}$ and $V_{2}$, respectively are
equivalent if there is an isomorphism $T:V_{1}\to V_{2}$ such that
$T\rho_{1}(g)=\rho_{2}(g)T$ for all $g\in\Gamma$.

Let $\rho:\Gamma\to GL(V)$ be a representation. The character
$\chi_{\rho}:\Gamma\to\mathbb{C}$ of $\rho$ is defined by setting
$\chi_{\rho}(g)=Tr(\rho(g))$ for $g\in\Gamma$, where $Tr(\rho(g))$ is the
trace of the matrix of $\rho(g)$ with respect to a basis of $V$. By the degree
of $\chi_{\rho}$, we mean the degree of $\rho$, which is simply
$\chi_{\rho}(1)$. If $W$ is a $\rho(g)$-invariant subspace of $V$ for each
$g\in\Gamma$, then we call $W$ a $\rho(\Gamma)$-invariant subspace of $V$. If
the $\rho(\Gamma)$-invariant subspaces of $V$ are $\\{0\\}$ and $V$ only, then
we call $\rho$ an irreducible representation of $\Gamma$, and the
corresponding character $\chi_{\rho}$ an irreducible character of $\Gamma$.

For a group $\Gamma$, we denote by $\text{IRR}(\Gamma)$ and
$\text{Irr}(\Gamma)$ the complete set of pairwise non-equivalent irreducible
representations of $\Gamma$ and the complete set of pairwise non-equivalent
irreducible characters of $\Gamma$, respectively. Note that
$\text{IRR}(\Gamma)$ is not necessarily unique, whereas $\text{Irr}(\Gamma)$
is unique.

For a representation $\rho:\Gamma\to GL(V)$ of a finite group $\Gamma$, define
$\overline{\rho}:\Gamma\to GL(V)$ by $\overline{\rho}(g)=\overline{\rho(g)}$,
where $\overline{\rho(g)}$ is the matrix whose entries are the complex
conjugates of the corresponding entries of $\rho(g)$. Note that if $\rho$ is
irreducible, then $\overline{\rho}$ is also irreducible. Hence we have the
following lemma. See Proposition 9.1.1 and Corollary 9.1.2 in [28] for
details.

######  Lemma 2.1.

Let $\Gamma$ be a finite group,
$\text{Irr}(\Gamma)=\\{\chi_{1},...,\chi_{h}\\}$ and $j\in\\{1,...,h\\}$. Then
there exists a $k\in\\{1,...,h\\}$ such that $\overline{\chi}_{k}=\chi_{j}$,
where $\overline{\chi}_{k}:\Gamma\to\mathbb{C}$ such that
$\overline{\chi}_{k}(x)=\overline{\chi_{k}(x)}$ for all $x\in\Gamma$.

Let $\Gamma$ be a finite group and $\alpha:\Gamma\to\mathbb{C}$ be a function.
The Cayley color digraph of $\Gamma$ with connection function $\alpha$,
denoted by $\text{Cay}(\Gamma,\alpha)$, is defined to be the graph (with
loops) having vertex set $\Gamma$ and edge set $\\{(x,y):x,y\in\Gamma\\}$ such
that each edge $(x,y)$ is colored by $\alpha(yx^{-1})$. The adjacency matrix
of $\text{Cay}(\Gamma,\alpha)$ is defined to be the matrix whose rows and
columns are indexed by the elements of $\Gamma$, and the $(x,y)$-entry is
equal to $\alpha(yx^{-1})$. The eigenvalues of $\text{Cay}(\Gamma,\alpha)$ are
simply the eigenvalues of its adjacency matrix.

Let $w_{m}=\exp(\frac{2\pi i}{m})$ be a primitive $m$-th root of unity. In the
special case when $\alpha:\Gamma\to\\{0,1,\omega_{6},\omega_{6}^{5}\\}$ with
$\alpha(g)=\overline{\alpha(g^{-1})}$ and the set $S=\\{g:\alpha(g)=1\text{ or
}\omega_{6}\\}$ satisfying $1\not\in S$ are chosen, the adjacency matrix of
$\text{Cay}(\Gamma,\alpha)$ is equal to the Hermitian-adjacency matrix of the
second kind of the mixed graph $\text{Cay}(\Gamma,S)$. Similarly, if
$\alpha:\Gamma\to\\{0,1\\}$ and the set $S=\\{g:\alpha(g)=1\\}$ satisfying
$1\not\in S$ are chosen, then the adjacency matrix of
$\text{Cay}(\Gamma,\alpha)$ is equal to the (0, 1)-adjacency matrix of the
mixed graph $\text{Cay}(\Gamma,S)$.

######  Theorem 2.2.

[11] Let $\alpha:\Gamma\to\mathbb{C}$ be a class function and
$\text{Irr}(\Gamma)=\\{\chi_{1},...,\chi_{h}\\}$. Then the spectrum of the
Cayley color digraph $\text{Cay}(\Gamma,\alpha)$ can be arranged as
$\\{[\lambda_{1}]^{d_{1}^{2}},...,[\lambda_{h}]^{d_{h}^{2}}\\},$ where

$\lambda_{j}=\frac{1}{\chi_{j}(1)}\sum_{x\in\Gamma}\alpha(x)\chi_{j}(x)\textnormal{
and }d_{j}=\chi_{j}(1)\textnormal{ for all }j=1,...,h.$

######  Lemma 2.3.

Let $\text{Cay}(\Gamma,S)$ be a normal mixed Cayley graph and
$\text{Irr}(\Gamma)=\\{\chi_{1},...,\chi_{h}\\}$. Then the HS-spectrum of
$\text{Cay}(\Gamma,S)$ can be arranged as
$\\{[\gamma_{1}]^{d_{1}^{2}},...,[\gamma_{h}]^{d_{h}^{2}}\\},$ where
$\gamma_{j}=\lambda_{j}+\mu_{j}$,

$\lambda_{j}=\frac{1}{\chi_{j}(1)}\sum_{s\in
S\setminus\overline{S}}\chi_{j}(s),\hskip
5.69046pt\mu_{j}=\frac{1}{\chi_{j}(1)}\sum_{s\in\overline{S}}(\omega_{6}\chi_{j}(s)+\omega_{6}^{5}\chi_{j}(s^{-1}))\textnormal{
and }d_{j}=\chi_{j}(1)\textnormal{ for all }j=1,...,h.$

###### Proof.

Let $\alpha:\Gamma\to\\{0,1,\omega_{6},\omega_{6}^{5}\\}$ be the function such
that

$\alpha(s)=\left\\{\begin{array}[]{rl}1&\mbox{if }s\in
S\setminus\overline{S}\\\ \omega_{6}&\mbox{if }s\in\overline{S}\\\
\omega_{6}^{5}&\mbox{if }s\in\overline{S}^{-1}\\\
0&\mbox{otherwise}.\end{array}\right.$

Since $S$ is a union of some conjugacy classes of $\Gamma$, $\alpha$ is a
class function. By Theorem 2.2,

$\gamma_{j}=\frac{1}{\chi_{j}(1)}\bigg{(}\sum_{s\in
S\setminus\overline{S}}\chi_{j}(s)+\sum_{s\in\overline{S}}\omega_{6}\chi_{j}(s)+\sum_{s\in\overline{S}^{-1}}\omega_{6}^{5}\chi_{j}(s)\bigg{)},$

and the result follows. ∎

As special cases of Lemma 2.3, we have the following two corollaries.

######  Corollary 2.3.1.

Let $\text{Cay}(\Gamma,S)$ be a normal simple Cayley graph and
$\text{Irr}(\Gamma)=\\{\chi_{1},...,\chi_{h}\\}$. Then the spectrum (or HS-
spectrum) of $\text{Cay}(\Gamma,S)$ can be arranged as
$\\{[\lambda_{1}]^{d_{1}^{2}},...,[\lambda_{h}]^{d_{h}^{2}}\\},$ where

$\lambda_{j}=\frac{1}{\chi_{j}(1)}\sum_{s\in S}\chi_{j}(s)\textnormal{ and
}d_{j}=\chi_{j}(1)\hskip 5.69046pt\textnormal{ for all }j=1,...,h.$

######  Corollary 2.3.2.

Let $\text{Cay}(\Gamma,S)$ be a normal oriented Cayley graph and
$\text{Irr}(\Gamma)=\\{\chi_{1},...,\chi_{h}\\}$. Then the HS-spectrum of
$\text{Cay}(\Gamma,S)$ can be arranged as
$\\{[\mu_{1}]^{d_{1}^{2}},...,[\mu_{h}]^{d_{h}^{2}}\\},$ where

$\mu_{j}=\frac{1}{\chi_{j}(1)}\sum_{s\in
S}(\omega_{6}\chi_{j}(s)+\omega_{6}^{5}\chi_{j}(s^{-1}))\textnormal{ and
}d_{j}=\chi_{j}(1)\hskip 5.69046pt\textnormal{ for all }j=1,...,h.$

Let $n\geq 2$ be a positive integer. For a divisor $d$ of $n$, define
$G_{n}(d)=\\{k:1\leq k\leq n-1,\gcd(k,n)=d\\}$. It is clear that
$G_{n}(d)=dG_{\frac{n}{d}}(1)$.

Alperin and Peterson [3] considered the Boolean algebra generated by the class
of subgroups of a group in order to determine the integrality of simple Cayley
graphs over abelian groups. Suppose $\mathcal{F}_{\Gamma}$ is the family of
all subgroups of $\Gamma$. The Boolean algebra $\mathbb{B}(\Gamma)$ generated
by $\mathcal{F}_{\Gamma}$ is the set whose elements are obtained by arbitrary
finite intersections, unions, and complements of the elements in the family
$\mathcal{F}_{\Gamma}$. The minimal non-empty elements of this algebra are
called atoms. Thus each element of $\mathbb{B}(\Gamma)$ is the union of some
atoms. Consider the equivalence relation $\sim$ on $\Gamma$ such that $x\sim
y$ if and only if $y=x^{k}$ for some $k\in G_{m}(1)$, where $m=\text{ord}(x)$.

######  Lemma 2.4.

[3] The equivalence classes of $\sim$ are the atoms of $\mathbb{B}(\Gamma)$.

For $x\in\Gamma$, let $[x]$ denote the equivalence class of $x$ with respect
to the relation $\sim$. Also, let $\langle x\rangle$ denote the cyclic
subgroup generated by $x$.

######  Lemma 2.5.

[3] The atoms of the Boolean algebra $\mathbb{B}(\Gamma)$ are the sets
$[x]=\\{y:\langle y\rangle=\langle x\rangle\\}$.

By Lemma 2.5, each element of $\mathbb{B}(\Gamma)$ is a union of some sets of
the form $[x]=\\{y:\langle y\rangle=\langle x\rangle\\}$. Thus for all
$S\in\mathbb{B}(\Gamma)$, we have $S=[x_{1}]\cup...\cup[x_{k}]$ for some
$x_{1},...,x_{k}\in\Gamma$.

######  Theorem 2.6.

[12] Let $\text{Cay}(\Gamma,S)$ be a normal simple Cayley graph. Then
$\text{Cay}(\Gamma,S)$ is integral if and only if $S\in\mathbb{B}(\Gamma)$.

Let $n\equiv 0\pmod{3}$. For a divisor $d$ of $\frac{n}{3}$ and
$r\in\\{1,2\\}$, define

$G_{n,3}^{r}(d)=\\{dk:k\equiv r\ (\mathrm{mod}\ 3),\gcd(dk,n)=d\\}.$

It is easy to see that $G_{n}(d)=G_{n,3}^{1}(d)\cup G_{n,3}^{2}(d)$ is a
disjoint union and $G_{n,3}^{r}(d)=dG_{\frac{n}{d}}^{r}(1)$ for $r=1,2$.

Define $\Gamma(3)$ to be the set of all $x\in\Gamma$ satisfying
$\text{ord}(x)\equiv 0\pmod{3}$. Define an equivalence relation $\approx$ on
$\Gamma(3)$ such that $x\approx y$ if and only if $y=x^{k}$ for some $k\in
G_{m,3}^{1}(1)$, where $m=\text{ord}(x)$. Observe that if $x,y\in\Gamma(3)$
and $x\approx y$ then $x\sim y$, but the converse need not be true. For
example, consider $x=5\pmod{12}$, $y=7\pmod{12}$ in $\mathbb{Z}_{12}$. Here
$x,y\in\mathbb{Z}_{12}(3)$ and $x\sim y$ but $x\not\approx y$. For
$x\in\Gamma(3)$, let $\langle\\!\langle x\rangle\\!\rangle$ denote the
equivalence class of $x$ with respect to the relation $\approx$.

##  3 HS-integral normal oriented Cayley graphs

Let $\text{Irr}(\Gamma)=\\{\chi_{1},...,\chi_{h}\\}$. Let $E$ be the matrix of
size $h\times n$, whose rows are indexed by $1,...,h$ and columns are indexed
by the elements of $\Gamma$ such that $E_{j,g}=\chi_{j}(g)$. Note that
$EE^{*}=nI_{h}$ and the rank of $E$ is $h$, where $E^{*}$ is the conjugate
transpose of E.

It is well known that the Galois group
$Gal(\mathbb{Q}(w_{m})/\mathbb{Q})=\\{\sigma_{r}:r\in G_{m}(1)\\}$, where
$\sigma_{r}(w_{m})=w_{m}^{r}$. Assume that $m\equiv 0\ (\mathrm{mod}\ 3)$.
Since $\mathbb{Q}(\omega_{3},w_{m})=\mathbb{Q}(w_{m})$, the Galois group
$Gal(\mathbb{Q}(\omega_{3},w_{m})/\mathbb{Q}(\omega_{3}))$ is a subgroup of
$Gal(\mathbb{Q}(w_{m})/\mathbb{Q})$. Therefore,
$Gal(\mathbb{Q}(\omega_{3},w_{m})/\mathbb{Q}(\omega_{3}))$ contains all those
automorphisms from $Gal(\mathbb{Q}(w_{m})/\mathbb{Q})$ that fixes
$\omega_{3}$. Note that $G_{m}(1)=G_{m,3}^{1}(1)\cup G_{m,3}^{2}(1)$ is a
disjoint union. Using $\sigma_{r}(\omega_{3})=\omega_{3}$ for all $r\in
G_{m,3}^{1}(1)$ and $\sigma_{r}(\omega_{3})=\omega_{3}^{2}$ for all $r\in
G_{m,3}^{2}(1)$, we get

$Gal(\mathbb{Q}(\omega_{3},w_{m})/\mathbb{Q}(\omega_{3}))=Gal(\mathbb{Q}(w_{m})/\mathbb{Q}(\omega_{3}))=\\{\sigma_{r}:r\in
G_{m,3}^{1}(1)\\}.$

Now assume that $m\not\equiv 0\ (\mathrm{mod}\ 3)$. Then

$[\mathbb{Q}(\omega_{3},w_{m}):\mathbb{Q}(\omega_{3})]=\frac{[\mathbb{Q}(\omega_{3},w_{m}):\mathbb{Q}(w_{m})]\times[\mathbb{Q}(w_{m}):\mathbb{Q}]}{[\mathbb{Q}(\omega_{3}):\mathbb{Q}]}=[\mathbb{Q}(w_{m}):\mathbb{Q}]=\varphi(m).$

Thus the field $\mathbb{Q}(\omega_{3},w_{m})$ is a Galois extension of
$\mathbb{Q}(\omega_{3})$ of degree $\varphi(m)$. Here $\varphi$ denotes the
Euler $\varphi$-function. Any automorphism of the field
$\mathbb{Q}(\omega_{3},w_{m})$ is uniquely determined by its action on
$w_{m}$. Hence
$Gal(\mathbb{Q}(\omega_{3},w_{m})/\mathbb{Q}(\omega_{3}))=\\{\tau_{r}:r\in
G_{m}(1)\\}$, where $\tau_{r}(w_{m})=w_{m}^{r}\text{ and
}\tau_{r}(\omega_{3})=\omega_{3}$. For $z\in\mathbb{C}$, let $\overline{z}$
denote the complex conjugate of $z$ and $\Re(z)$ (resp. $\Im(z)$) denote the
real part (resp. imaginary part) of the complex number $z$.

######  Theorem 3.1.

Let $\Gamma$ be a finite group,
$\text{Irr}(\Gamma)=\\{\chi_{1},...,\chi_{h}\\}$, and
$x=\sum\limits_{g\in\Gamma}c_{g}g$ an element in
$\mathbb{Q}(\omega_{3})\Gamma$. Then $\chi_{j}(x)$ is rational for each $1\leq
j\leq h$ if and only if the following conditions hold:

  1. (i)

$\sum\limits_{s\in Cl(g)}c_{s}=\sum\limits_{s\in Cl(g^{-1})}\overline{c_{s}}$
for all $g\in\Gamma$,

  2. (ii)

$\sum\limits_{s\in Cl(g_{1})}c_{s}=\sum\limits_{s\in Cl(g_{2})}c_{s}$ for all
$g_{1},g_{2}\in\Gamma(3)$ and $g_{1}\approx g_{2}$,

  3. (iii)

$\Re\left(\sum\limits_{s\in Cl(g)}c_{s}\right)=\Re\left(\sum\limits_{s\in
Cl(g^{-1})}c_{s}\right)$ for all $g\in\Gamma\setminus\Gamma(3)$, and

  4. (iv)

$\Im\left(\sum\limits_{s\in Cl(g)}c_{s}\right)=0$ for all
$g\in\Gamma\setminus\Gamma(3)$.

###### Proof.

Let $L$ be a set of representatives of the conjugacy classes in $\Gamma$.
Since characters are class functions, we have

$\displaystyle\chi_{j}(x)=\sum_{g\in L}\bigg{(}\sum_{s\in
Cl(g)}c_{s}\bigg{)}\chi_{j}(g)\textnormal{ for all }1\leq j\leq h.$ (1)

Assume that $\chi_{j}(x)$ is rational for each $1\leq j\leq h$. Let
$g_{1},g_{2}\in\Gamma(3)$, $g_{1}\approx g_{2}$ and $m=\text{ord}(g_{1})$.
Then there is $r\in G_{m,3}^{1}(1)$ and $\sigma_{r}\in
Gal(\mathbb{Q}(w_{m})/\mathbb{Q}(\omega_{3}))$ such that $g_{2}=g_{1}^{r}$ and
$\sigma_{r}(w_{m})=w_{m}^{r}$. Note that
$\sigma_{r}(\chi_{j}(g_{1}))=\chi_{j}(g_{1}^{r})$ for all $1\leq j\leq h$. For
$t\in\Gamma$, let
$\theta_{t}=\sum\limits_{j=1}^{h}\chi_{j}(t)\overline{\chi}_{j}$, where
$\overline{\chi}_{j}(g)=\overline{\chi_{j}(g)}$ for all $g\in\Gamma$. By
orthogonality of characters, we have

$\displaystyle\theta_{t}(u)=\left\\{\begin{array}[]{rl}|C_{\Gamma}(t)|&\mbox{
if $u$ and $t$ are conjugates to each other}\\\ 0&\mbox{
otherwise}.\end{array}\right.$ (4)

So $\theta_{t}(x)=|C_{\Gamma}(t)|\sum\limits_{s\in
Cl(t)}c_{s}\in\mathbb{Q}(\omega_{3})$ implies that
$\sigma_{r}(\theta_{t}(x))=\theta_{t}(x)$. By assumption,
$\sigma_{r}(\chi_{j}(x))=\chi_{j}(x)$ for all $j=1,...,h$. Thus

$\displaystyle|C_{\Gamma}(g_{1})|\sum\limits_{s\in
Cl(g_{1})}c_{s}=\theta_{g_{1}}(x)=\sigma_{r}(\theta_{g_{1}}(x))$
$\displaystyle=\sum\limits_{j=1}^{h}\sigma_{r}(\chi_{j}(g_{1}))\sigma_{r}(\overline{\chi}_{j}(x))$
$\displaystyle=\sum\limits_{j=1}^{h}\chi_{j}(g_{1}^{r})\overline{\chi}_{j}(x)$
$\displaystyle=\theta_{g_{1}^{r}}(x)=\theta_{g_{2}}(x)=|C_{\Gamma}(g_{2})|\sum\limits_{s\in
Cl(g_{2})}c_{s}.$ (5)

Since $g_{1}\approx g_{2}$, we have $C_{\Gamma}(g_{1})=C_{\Gamma}(g_{2})$. So
Equation $(\ref{ConjEqThetaEqua})$ implies that $\sum\limits_{s\in
Cl(g_{1})}c_{s}=\sum\limits_{s\in Cl(g_{2})}c_{s}$. Hence condition $(ii)$
holds.

Again

$\displaystyle 0=\chi_{j}(x)-\overline{\chi_{j}(x)}$ $\displaystyle=\sum_{g\in
L}\bigg{(}\sum_{s\in Cl(g)}c_{s}\bigg{)}\chi_{j}(g)-\sum_{g\in
L}\bigg{(}\sum_{s\in Cl(g)}\overline{c_{s}}\bigg{)}\overline{\chi_{j}(g)}$
$\displaystyle=\sum_{g\in L}\bigg{(}\sum_{s\in
Cl(g)}c_{s}\bigg{)}\chi_{j}(g)-\sum_{g\in L}\bigg{(}\sum_{s\in
Cl(g)}\overline{c_{s}}\bigg{)}\chi_{j}(g^{-1})$ $\displaystyle=\sum_{g\in
L}\bigg{(}\sum_{s\in Cl(g)}c_{s}-\sum_{s\in
Cl(g^{-1})}\overline{c_{s}}\bigg{)}\chi_{j}(g),$

and so

$\displaystyle\sum_{g\in L}\bigg{(}\sum_{s\in Cl(g)}c_{s}-\sum_{s\in
Cl(g^{-1})}\overline{c_{s}}\bigg{)}\begin{bmatrix}\chi_{1}(g)\\\
\chi_{2}(g)\\\ \vdots\\\ \chi_{h}(g)\end{bmatrix}=\begin{bmatrix}0\\\ 0\\\
\vdots\\\ 0\end{bmatrix}.$ (6)

Since characters are class functions and the rank of $E$ is $h$, the columns
of $E$ corresponding to the elements of $L$ are linearly independent. Thus by
Equation $(\ref{eqLIsumZero})$, $\bigg{(}\sum\limits_{s\in
Cl(g)}c_{s}-\sum\limits_{s\in Cl(g^{-1})}\overline{c_{s}}\bigg{)}=0$ for all
$g\in L$, and so condition $(i)$ holds.

Let $g\in\Gamma\setminus\Gamma(3)$ and $m=\text{ord}(g)$. Then there exists
$\tau_{m-1}\in Gal(\mathbb{Q}(\omega_{3},w_{m})/\mathbb{Q}(\omega_{3}))$ such
that $\tau_{m-1}(w_{m})=w_{m}^{m-1}$. Note that
$\tau_{m-1}(\chi_{j}(g))=\chi_{j}(g^{m-1})$ for all $1\leq j\leq h$. Now

$\displaystyle|C_{\Gamma}(g)|\sum\limits_{s\in
Cl(g)}c_{s}=\theta_{g}(x)=\tau_{m-1}(\theta_{g}(x))$
$\displaystyle=\sum\limits_{j=1}^{h}\tau_{m-1}(\chi_{j}(g))\tau_{m-1}(\overline{\chi}_{j}(x))$
$\displaystyle=\sum\limits_{j=1}^{h}\chi_{j}(g^{m-1})\overline{\chi}_{j}(x)$
$\displaystyle=\theta_{g^{m-1}}(x)=\theta_{g^{-1}}(x)=|C_{\Gamma}(g^{-1})|\sum\limits_{s\in
Cl(g^{-1})}c_{s}.$ (7)

Since $C_{\Gamma}(g)=C_{\Gamma}(g^{-1})$, Equation $(\ref{ConjEqThetaEqua2})$
implies that $\sum\limits_{s\in Cl(g)}c_{s}=\sum\limits_{s\in
Cl(g^{-1})}c_{s}$. This, together with condition $(i)$, imply that condition
$(iii)$ and $(iv)$ hold.

Conversely, assume that all the given conditions hold. Let
$n=\text{ord}(\Gamma)$. We have the following two cases.

Case 1. Assume that $n\equiv 0\ (\mathrm{mod}\ 3)$. Let $\sigma_{k}\in
Gal(\mathbb{Q}(\omega_{3},w_{n})/\mathbb{Q}(\omega_{3}))$. Then
$\sigma_{k}(w_{n})=w_{n}^{k}$ and $k\in G_{n,3}^{1}(1)$, and so
$\sigma_{k}(\chi_{j}(g))=\chi_{j}(g^{k})$ for all $1\leq j\leq h$. Thus

$\displaystyle\sigma_{k}(\chi_{j}(x))$ $\displaystyle=\sum_{g\in
L}\bigg{(}\sum_{s\in Cl(g)}c_{s}\bigg{)}\sigma_{k}(\chi_{j}(g))$
$\displaystyle=\sum_{g\in L(3)}\bigg{(}\sum_{s\in
Cl(g)}c_{s}\bigg{)}\chi_{j}(g^{k}).$ (8)

Since $g\approx g^{k}$, by condition $(ii)$ we have $\sum\limits_{s\in
Cl(g)}c_{s}=\sum\limits_{s\in Cl(g^{k})}c_{s}$. From Equation
$(\ref{ConjEqThetaEqua3})$, we get

$\displaystyle\sigma_{k}(\chi_{j}(x))$ $\displaystyle=\sum_{g\in
L}\bigg{(}\sum_{s\in Cl(g^{k})}c_{s}\bigg{)}\chi_{j}(g^{k})=\chi_{j}(x).$ (9)

The second equality in $(\ref{neweq})$ holds, because $\\{g^{k}:g\in L\\}$ is
also a set of representatives of conjugacy classes of $\Gamma$. Now
$\sigma_{k}(\chi_{j}(x))=\chi_{j}(x)$ for all $k\in G_{n,3}^{1}(1)$ implies
that $\chi_{j}(x)\in\mathbb{Q}(\omega_{3})$.

Case 2. Assume that $n\not\equiv 0\ (\mathrm{mod}\ 3)$. As in Case 1, we get
$\tau_{r}(\chi_{j}(x))=\chi_{j}(x)$ for all $r\in G_{n}(1)$, where
$\tau_{r}\in Gal(\mathbb{Q}(\omega_{3},w_{m})/\mathbb{Q}(\omega_{3}))$. Thus
$\chi_{j}(x)\in\mathbb{Q}(\omega_{3})$.
In both cases, we get $\chi_{j}(x)\in\mathbb{Q}(\omega_{3})$. Taking complex
conjugates in Equation $(\ref{ConjEqThetaEqua5})$,

$\displaystyle\overline{\chi_{j}(x)}=\sum_{g\in L}\bigg{(}\sum_{s\in
Cl(g)}\overline{c_{s}}\bigg{)}\overline{\chi_{j}(g)}$
$\displaystyle=\sum_{g\in L}\bigg{(}\sum_{s\in
Cl(g)}\overline{c_{s}}\bigg{)}\chi_{j}(g^{-1})$ $\displaystyle=\sum_{g\in
L}\bigg{(}\sum_{s\in Cl(g^{-1})}c_{s}\bigg{)}\chi_{j}(g^{-1})$
$\displaystyle=\chi_{j}(x).$ (10)

Equation $(\ref{ConjEqThetaEqua4})$ implies that $\chi_{j}(x)\in\mathbb{Q}$. ∎

For $\Gamma(3)=\emptyset$, define $\mathbb{E}(\Gamma)=\\{\emptyset\\}$. For
$\Gamma(3)\neq\emptyset$, define $\mathbb{E}(\Gamma)$ to be the set of all
skew-symmetric subsets $S$, where $S=\langle\\!\langle
x_{1}\rangle\\!\rangle\cup...\cup\langle\\!\langle
x_{k}\rangle\\!\rangle=Cl(y_{1})\cup...\cup Cl(y_{l})$ for some
$x_{1},...,x_{k}\in\Gamma(3)$ and $y_{1},...,y_{l}\in\Gamma$. We can replace
condition $(ii)$ of Theorem 3.1 by $\sum\limits_{s\in
Cl(x)}c_{s}=\sum\limits_{s\in Cl(y)}c_{s}$ for all $x,y\in\langle\\!\langle
g\rangle\\!\rangle$ and $g\in\Gamma(3)$.

######  Theorem 3.2.

Let $\text{Cay}(\Gamma,S)$ be a normal oriented Cayley graph. Then
$\text{Cay}(\Gamma,S)$ is HS-integral if and only if $S\in\mathbb{E}(\Gamma)$.

###### Proof.

Let $\text{Irr}(\Gamma)=\\{\chi_{1},...,\chi_{h}\\}$ and
$x=\sum\limits_{g\in\Gamma}c_{g}g$, where

$c_{g}=\left\\{\begin{array}[]{rl}-\omega_{3}^{2}&\mbox{if }g\in S\\\
-\omega_{3}&\mbox{if }g\in S^{-1}\\\ 0&\mbox{otherwise}.\end{array}\right.$

Observe that $\chi_{j}(x)=\sum\limits_{s\in
S}(-\omega_{3}^{2}\chi_{j}(s)-\omega_{3}\chi_{j}(s^{-1}))$, so that
$\frac{\chi_{j}(x)}{\chi_{j}(1)}$ is an HS-eigenvalue of
$\text{Cay}(\Gamma,S)$. Assume that the normal oriented Cayley graph
$\text{Cay}(\Gamma,S)$ is HS-integral. Here $S$ is the union of some conjugacy
classes of $\Gamma$. Then $\chi_{j}(x)$ is an integer for each $1\leq j\leq
h$, and therefore all the four conditions of Theorem 3.1 hold. By the fourth
condition of Theorem 3.1, we get $\Im\left(\sum\limits_{s\in
Cl(g)}c_{s}\right)=0$ for all $g\in\Gamma\setminus\Gamma(3)$. If $g\in S$,
then $Cl(g)\subseteq S$, and so by the definition of $c_{g}$, we get
$\Im\left(\sum\limits_{s\in Cl(g)}c_{s}\right)=\frac{\sqrt{3}|Cl(g)|}{2}\neq
0$. Thus $S\cap(\Gamma\setminus\Gamma(3))=\emptyset$, that is,
$S\subseteq\Gamma(3)$. Again, let $g_{1}\in S$, $g_{2}\in\Gamma(3)$ and
$g_{1}\approx g_{2}$. By the second condition of Theorem 3.1, we get
$0\neq\sum\limits_{s\in Cl(g_{1})}c_{s}=\sum\limits_{s\in Cl(g_{2})}c_{s}$,
which implies that $g_{2}\in S$. Thus $g_{1}\in S$ gives $\langle\\!\langle
g_{1}\rangle\\!\rangle\subseteq S$. Hence $S\in\mathbb{E}(\Gamma)$.

Conversely, assume that $S\in\mathbb{E}(\Gamma)$. Let $S=\langle\\!\langle
x_{1}\rangle\\!\rangle\cup...\cup\langle\\!\langle
x_{r}\rangle\\!\rangle=Cl(y_{1})\cup...\cup Cl(y_{k})\subseteq\Gamma(3)$ for
some $x_{1},...,x_{r}\in\Gamma(3)$ and $y_{1},...,y_{k}\in\Gamma$. Then
$S^{-1}=\langle\\!\langle
x_{1}^{-1}\rangle\\!\rangle\cup...\cup\langle\\!\langle
x_{r}^{-1}\rangle\\!\rangle=Cl(y_{1}^{-1})\cup...\cup
Cl(y_{k}^{-1})\subseteq\Gamma(3)$. For $g_{1},g_{2}\in\Gamma(3)$, if
$g_{1}\approx g_{2}$ then $Cl(g_{1}),Cl(g_{2})\subseteq S$ or
$Cl(g_{1}),Cl(g_{2})\subseteq S^{-1}$ or $Cl(g_{1}),Cl(g_{2})\subseteq(S\cup
S^{-1})^{c}$. Note that $|Cl(g_{1})|=|Cl(g_{2})|$. For all the cases, using
the definition of $c_{g}$, we find

$\sum\limits_{s\in Cl(g_{1})}c_{s}=\sum\limits_{s\in Cl(g_{2})}c_{s}.$

Thus condition $(ii)$ of Theorem 3.1 holds. Again for $g\in\Gamma$,
$Cl(g)\subseteq S$ if and only if $Cl(g^{-1})\subseteq S^{-1}$. Therefore
$\sum\limits_{s\in Cl(g)}c_{s}=\sum\limits_{s\in Cl(g^{-1})}\overline{c_{s}}$,
and so condition $(i)$ of Theorem 3.1 also holds. Further, if
$g\not\in\Gamma(3)$, then $Cl(g)\cap(S\cup S^{-1})=Cl(g^{-1})\cap(S\cup
S^{-1})=\emptyset$, and so $\sum\limits_{s\in Cl(g)}c_{s}=0=\sum\limits_{s\in
Cl(g^{-1})}c_{s}$, therefore condition $(iii)$ and $(iv)$ of Theorem 3.1 also
hold. Thus by Theorem 3.1, $\chi_{j}(x)$ is rational for each $1\leq j\leq h$.
Consequently, the HS-eigenvalue $\mu_{j}=\frac{\chi_{j}(x)}{\chi_{j}(1)}$ of
the normal oriented Cayley graph $\text{Cay}(\Gamma,S)$ is a rational
algebraic integer, and hence an integer for each $1\leq j\leq h$. ∎

$I$$(1,2)(3,4)$$(1,3)(2,4)$$(1,4)(2,3)$$(1,2,3)$$(4,2,1)$$(2,4,3)$$(3,4,1)$$(1,3,2)$$(4,1,2)$$(2,3,4)$$(3,1,4)$
Figure 1:  The oriented graph
$\text{Cay}(A_{4},\\{(1,2,3),(4,2,1),(2,4,3),(3,4,1)\\})$

######  Example 3.1.

Consider the alternating group $A_{4}$ and
$S=\\{(1,2,3),(4,2,1),(2,4,3),(3,4,1)\\}$. The conjugacy classes of $A_{4}$
are $\\{I\\},\\{(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\\},$
$\\{(1,2,3),(4,2,1),(2,4,3),(3,4,1)\\}$ and
$\\{(1,3,2),(4,1,2),(2,3,4),(3,1,4)\\}$. The normal oriented Cayley graph
$\text{Cay}(A_{4},S)$ is shown in Figure 1. We see that
$S=\langle\\!\langle(1,2,3)\rangle\\!\rangle\cup\langle\\!\langle(4,2,1)\rangle\\!\rangle\cup\langle\\!\langle(2,4,3)\rangle\\!\rangle\cup\langle\\!\langle(3,4,1)\rangle\\!\rangle=Cl((1,2,3))$.
Therefore $S\in\mathbb{E}(\Gamma)$. From [15], the character table of $A_{4}$
is given in Table 1, where
$Irr(A_{4})=\\{\chi_{1},\chi_{2},\chi_{3},\chi_{4}\\}$. Further, using
Corollary 2.3.2, the HS-spectrum of $\text{Cay}(A_{4},S)$ is obtained as
$\\{[\mu_{1}]^{1},[\mu_{2}]^{1},[\mu_{3}]^{1},[\mu_{4}]^{9}\\},$ where
$\mu_{1}=4(\omega_{6}+\omega_{6}^{5})=4$,
$\mu_{2}=4(\omega_{6}\omega_{3}+\omega_{6}^{5}\omega_{3}^{2})=-8$,
$\mu_{3}=4(\omega_{6}\omega_{3}^{2}+\omega_{6}^{5}\omega_{3})=4$ and
$\mu_{4}=0$. Thus $\text{Cay}(A_{4},S)$ is HS-integral.

Table 1: Character table of $A_{4}$ |  $I$ |  $(1,2)(3,4)$ |  (1,2,3) |  (1,3,2)
---|---|---|---|---
$\chi_{1}$ |  $1$ |  $~{}~{}1$ |  $1$ |  $1$
$\chi_{2}$ |  $1$ |  $~{}~{}1$ |  $\omega_{3}$ |  $\omega_{3}^{2}$
$\chi_{3}$ |  $1$ |  $~{}~{}1$ |  $\omega_{3}^{2}$ |  $\omega_{3}$
$\chi_{4}$ |  $3$ |  $-1$ |  $0$ |  $0$

##  4 HS-integral normal mixed Cayley graphs

######  Lemma 4.1.

Let $S$ be a skew-symmetric subset a finite group $\Gamma$ and that $S$ is the
union of some conjugacy classes of $\Gamma$. Let
$\text{Irr}(\Gamma)=\\{\chi_{1},...,\chi_{h}\\}$ and $t(\neq 0)\in\mathbb{Q}$.
If $\frac{1}{\chi_{j}(1)}\sum\limits_{s\in
S}t\left(i\sqrt{3}\chi_{j}(s)-i\sqrt{3}\chi_{j}(s^{-1})\right)$ is an integer
for each $1\leq j\leq h$ then $S\in\mathbb{E}(\Gamma)$

###### Proof.

Let $x=\sum\limits_{g\in\Gamma}c_{g}g\in\mathbb{Q}(\omega_{3})\Gamma$, where

$c_{g}=\left\\{\begin{array}[]{rl}it\sqrt{3}&\mbox{if }g\in S\\\
-it\sqrt{3}&\mbox{if }g\in S^{-1}\\\ 0&\mbox{otherwise}.\end{array}\right.$

Assume that
$\frac{\chi_{j}(x)}{\chi_{j}(1)}=\frac{1}{\chi_{j}(1)}\sum\limits_{s\in
S}t\left(i\sqrt{3}\chi_{j}(s)-i\sqrt{3}\chi_{j}(s^{-1})\right)$ is an integer
for each $1\leq j\leq h$. Then all the four conditions of Theorem 3.1 hold. By
the fourth condition of Theorem 3.1, we get $\Im\left(\sum\limits_{s\in
Cl(g)}c_{s}\right)=0$ for all $g\in\Gamma\setminus\Gamma(3)$, and so we must
have $S\cup S^{-1}\subseteq\Gamma(3)$. Again, let $g_{1}\in S$,
$g_{2}\in\Gamma(3)$ and $g_{1}\approx g_{2}$. The second condition of Theorem
3.1 gives $\sum\limits_{s\in Cl(g_{1})}c_{s}=\sum\limits_{s\in
Cl(g_{2})}c_{s}$. Note that $\sum\limits_{s\in
Cl(g_{1})}c_{s}=it\sqrt{3}|Cl(g_{1})|$. Therefore $\sum\limits_{s\in
Cl(g_{2})}c_{s}=it\sqrt{3}|Cl(g_{1})|$, and so $g_{2}\in S$. Thus $g_{1}\in S$
implies $\langle\\!\langle g_{1}\rangle\\!\rangle\subseteq S$. Hence
$S\in\mathbb{E}(\Gamma)$. ∎

######  Lemma 4.2.

Let $S$ be a skew-symmetric subset a finite group $\Gamma$ and that $S$ is the
union of some conjugacy classes of $\Gamma$. Let
$\text{Irr}(\Gamma)=\\{\chi_{1},...,\chi_{h}\\}$ and $t(\neq 0)\in\mathbb{Q}$.
If $\frac{1}{\chi_{j}(1)}\sum\limits_{s\in
S}t\left(i\sqrt{3}\chi_{j}(s)-i\sqrt{3}\chi_{j}(s^{-1})\right)$ is an integer
for each $1\leq j\leq h$ then $\frac{1}{\chi_{j}(1)}\sum\limits_{s\in S\cup
S^{-1}}\chi_{j}(s)$ is also an integer for each $1\leq j\leq h$.

###### Proof.

Assume that $\frac{1}{\chi_{j}(1)}\sum\limits_{s\in
S}t\left(i\sqrt{3}\chi_{j}(s)-i\sqrt{3}\chi_{j}(s^{-1})\right)$ is an integer
for each $1\leq j\leq h$. By Lemma 4.1 we have $S\in\mathbb{E}(\Gamma)$, and
so $S=\langle\\!\langle x_{1}\rangle\\!\rangle\cup...\cup\langle\\!\langle
x_{k}\rangle\\!\rangle$ for some $x_{1},...,x_{k}\in\Gamma(3)$. Therefore, we
get

$S\cup S^{-1}=\left(\langle\\!\langle
x_{1}\rangle\\!\rangle\cup...\cup\langle\\!\langle
x_{k}\rangle\\!\rangle\right)\cup\left(\langle\\!\langle
x_{1}^{-1}\rangle\\!\rangle\cup...\cup\langle\\!\langle
x_{k}^{-1}\rangle\\!\rangle\right)=[x_{1}]\cup...\cup[x_{k}]\in\mathbb{B}(\Gamma).$

Thus by Theorem 2.6, $\text{Cay}(\Gamma,S\cup S^{-1})$ is integral, that is,
$\frac{1}{\chi_{j}(1)}\sum\limits_{s\in S\cup S^{-1}}\chi_{j}(s)$ is an
integer for each $1\leq j\leq h$. ∎

In the next result, we will be using the fact that the HS-eigenvalues of a
mixed Cayley graph are rational integers. See Theorem 2.6 of [23] for details.

######  Lemma 4.3.

Let $\text{Cay}(\Gamma,S)$ be a normal mixed Cayley graph. Then
$\text{Cay}(\Gamma,S)$ is HS-integral if and only if
$\text{Cay}(\Gamma,{S\setminus\overline{S}})$ and
$\text{Cay}(\Gamma,{\overline{S}})$ are HS-integral.

###### Proof.

Let $\text{Irr}(\Gamma)=\\{\chi_{1},...,\chi_{h}\\}$. By Lemma 2.3, the HS-
spectrum of $\text{Cay}(\Gamma,S)$ is
$\\{[\gamma_{1}]^{d_{1}^{2}},...,[\gamma_{h}]^{d_{h}^{2}}\\},$ where
$\gamma_{j}=\lambda_{j}+\mu_{j}$,

$\lambda_{j}=\frac{1}{\chi_{j}(1)}\sum_{s\in
S\setminus\overline{S}}\chi_{j}(s),\hskip
5.69046pt\mu_{j}=\frac{1}{\chi_{j}(1)}\sum_{s\in\overline{S}}(\omega_{6}\chi_{j}(s)+\omega_{6}^{5}\chi_{j}(s^{-1}))\textnormal{
and }d_{j}=\chi_{j}(1)\textnormal{ for all }j=1,...,h.$

Note that $\\{[\lambda_{1}]^{d_{1}^{2}},...,[\lambda_{h}]^{d_{h}^{2}}\\}$ is
the spectrum of $\text{Cay}(\Gamma,S\setminus\overline{S})$ and
$\\{[\mu_{1}]^{d_{1}^{2}},...,[\mu_{h}]^{d_{h}^{2}}\\}$ is the HS-spectrum of
$\text{Cay}(\Gamma,\overline{S})$.

Assume that the mixed Cayley graph $\text{Cay}(\Gamma,S)$ is HS-integral. Let
$j\in\\{1,...,h\\}$. By Lemma 2.1, there exists $k\in\\{1,...,h\\}$ such that
$\chi_{k}=\overline{\chi}_{j}$. Then $\lambda_{j}=\lambda_{k}$ and
$\chi_{j}(1)=\chi_{k}(1)$. Thus

$\displaystyle\gamma_{j}-\gamma_{k}$
$\displaystyle=\frac{1}{\chi_{j}(1)}\sum_{s\in\overline{S}}\left(\omega_{6}\chi_{j}(s)+\omega_{6}^{5}\chi_{j}(s^{-1})\right)-\frac{1}{\chi_{k}(1)}\sum_{s\in\overline{S}}\left(\omega_{6}\chi_{k}(s)+\omega_{6}^{5}\chi_{k}(s^{-1})\right)$
$\displaystyle=\frac{1}{\chi_{j}(1)}\sum_{s\in\overline{S}}\left(\omega_{6}\chi_{j}(s)+\omega_{6}^{5}\chi_{j}(s^{-1})\right)-\frac{1}{\chi_{j}(1)}\sum_{s\in\overline{S}}\left(\omega_{6}\overline{\chi_{j}(s)}+\omega_{6}^{5}\overline{\chi_{j}(s^{-1})}\right)$
$\displaystyle=\frac{1}{\chi_{j}(1)}\sum_{s\in\overline{S}}\left(\omega_{6}\chi_{j}(s)+\omega_{6}^{5}\chi_{j}(s^{-1})\right)-\frac{1}{\chi_{j}(1)}\sum_{s\in\overline{S}}\left(\omega_{6}\chi_{j}(s^{-1})+\omega_{6}^{5}\chi_{j}(s)\right)$
$\displaystyle=\frac{1}{\chi_{j}(1)}\sum_{s\in\overline{S}}(\omega_{6}-\omega_{6}^{5})\chi_{j}(s)+(\omega_{6}^{5}-\omega_{6})\chi_{j}(s^{-1})$
$\displaystyle=\frac{1}{\chi_{j}(1)}\sum_{s\in\overline{S}}\left(i\sqrt{3}\chi_{j}(s)-i\sqrt{3}\chi_{j}(s^{-1})\right).$

By assumption $\gamma_{j},\gamma_{k}\in\mathbb{Z}$, and so
$\frac{1}{\chi_{j}(1)}\sum\limits_{s\in\overline{S}}\left(i\sqrt{3}\chi_{j}(s)-i\sqrt{3}\chi_{j}(s^{-1})\right)\in\mathbb{Z}$
for all $1\leq j\leq h$. By Lemma 4.2, we get
$\frac{1}{\chi_{j}(1)}\sum\limits_{s\in\overline{S}\cup\overline{S}^{-1}}\chi_{j}(s)\in\mathbb{Z}$
for all $1\leq j\leq h$. Since

$\displaystyle\mu_{j}=\frac{1}{2\chi_{j}(1)}\sum\limits_{s\in\overline{S}\cup\overline{S}^{-1}}\chi_{j}(s)+\frac{1}{2\chi_{j}(1)}\sum\limits_{s\in\overline{S}}\left(i\sqrt{3}\chi_{j}(s)-i\sqrt{3}\chi_{j}(s^{-1})\right),$

$\mu_{j}$ is a rational algebraic integer, and hence it is an integer for each
$1\leq j\leq h$. Thus $\text{Cay}(\Gamma,\overline{S})$ is HS-integral. Now we
have $\gamma_{j},\mu_{j}\in\mathbb{Z}$, and so
$\lambda_{j}=\gamma_{j}-\mu_{j}\in\mathbb{Z}$ for each $1\leq j\leq h$. Hence
$\text{Cay}(\Gamma,S\setminus\overline{S})$ is also HS-integral.

Conversely, assume that $\text{Cay}(\Gamma,S\setminus\overline{S})$ and
$\text{Cay}(\Gamma,\overline{S})$ are HS-integral. Then Lemma 2.3 implies that
$\text{Cay}(\Gamma,S)$ is integral. ∎

######  Theorem 4.4.

Let $\text{Cay}(\Gamma,S)$ be a normal mixed Cayley graph. Then
$\text{Cay}(\Gamma,S)$ is HS-integral if and only if
$S\setminus\overline{S}\in\mathbb{B}(\Gamma)$ and
$\overline{S}\in\mathbb{D}(\Gamma)$.

###### Proof.

By Lemma 4.3, $\text{Cay}(\Gamma,S)$ is HS-integral if and only if
$\text{Cay}(\Gamma,S\setminus\overline{S})$ and
$\text{Cay}(\Gamma,\overline{S})$ are HS-integral. Now the proof follows from
Theorem 2.6 and Theorem 3.2. ∎

$I$$(1,2)(3,4)$$(1,3)(2,4)$$(1,4)(2,3)$$(1,2,3)$$(4,2,1)$$(2,4,3)$$(3,4,1)$$(1,3,2)$$(4,1,2)$$(2,3,4)$$(3,1,4)$
Figure 2: The mixed graph
$\text{Cay}(A_{4},\\{(1,2)(3,4),(1,3)(2,4),(1,4)(2,3),(1,2,3),(4,2,1),(2,4,3),(3,4,1)\\})$

######  Example 4.1.

Consider the alternating group $A_{4}$ and

$S=\\{(1,2)(3,4),(1,3)(2,4),(1,4)(2,3),(1,2,3),(4,2,1),(2,4,3),(3,4,1)\\}.$

The normal mixed Cayley graph $\text{Cay}(A_{4},S)$ is shown in Figure 2. We
find that
$\overline{S}=\langle\\!\langle(1,2,3)\rangle\\!\rangle\cup\langle\\!\langle(4,2,1)\rangle\\!\rangle\cup\langle\\!\langle(2,4,3)\rangle\\!\rangle\cup\langle\\!\langle(3,4,1)\rangle\\!\rangle=Cl((1,2,3))\in\mathbb{E}(\Gamma)$,
and therefore we have
$S\setminus\overline{S}=[(1,2)(3,4)]\cup[(1,3)(2,4)]\cup[(1,4)(2,3)]=Cl((1,2)(3,4))\in\mathbb{B}(\Gamma)$.
The character table of $A_{4}$ is given in Table 1, where
$Irr(\Gamma)=\\{\chi_{1},\chi_{2},\chi_{3},\chi_{4}\\}$. Further, using Lemma
2.3, the HS-spectrum of $\text{Cay}(A_{4},S)$ is obtained as
$\\{[\gamma_{1}]^{1},[\gamma_{2}]^{1},[\gamma_{3}]^{1},[\gamma_{4}]^{9}\\},$
where $\gamma_{1}=3+4(\omega_{6}+\omega_{6}^{5})=7$,
$\gamma_{2}=3+4(\omega_{6}\omega_{3}+\omega_{6}^{5}\omega_{3}^{2})=-5$,
$\gamma_{3}=3+4(\omega_{6}\omega_{3}^{2}+\omega_{6}^{5}\omega_{3})=7$ and
$\gamma_{4}=-1$. Thus $\text{Cay}(A_{4},S)$ is HS-integral.

##  5 Eisenstein integral normal mixed Cayley graphs

Assume $S$ is the union of some conjugacy classes from the finite group
$\Gamma$ that doesn’t contain $1$ and
$Irr(\Gamma)=\\{\chi_{1},...,\chi_{h}\\}$. Consider the function
$\alpha:\Gamma\rightarrow\\{0,1\\}$ defined by

$\alpha(s)=\left\\{\begin{array}[]{rl}1&\mbox{if }s\in S\\\
0&\mbox{otherwise}.\end{array}\right.$

in Theorem 2.2. We observe that for each $1\leq j\leq h$,
$\frac{1}{\chi_{j}(1)}\sum\limits_{s\in S}\chi_{j}(s)$ is an eigenvalue of the
normal mixed Cayley graph $\text{Cay}(\Gamma,S)$.

For each $j\in\\{1,...,h\\}$, define

$f_{j}(S)=\frac{1}{\chi_{j}(1)}\sum_{s\in
S\setminus\overline{S}}\chi_{j}(s)\hskip 14.22636pt\textnormal{and}\hskip
14.22636ptg_{j}(S)=\frac{1}{\chi_{j}(1)}\sum_{s\in\overline{S}}(\omega\chi_{j}(s)+\overline{w}\chi_{j}(s^{-1})),$

where $\omega=\frac{1}{2}-\frac{\sqrt{3}i}{6}$.

Let $j\in\\{1,...,h\\}$. By Lemma 2.1, there exist $k\in\\{1,...,h\\}$, such
that $\chi_{k}=\overline{\chi}_{j}$. Now we have

$\begin{split}\frac{1}{\chi_{j}(1)}\sum_{s\in
S}\chi_{j}(s)=f_{j}(S)+g_{j}(S)+\omega_{3}(g_{j}(S)-g_{k}(S)).\end{split}$
(11)

Note that $f_{j}(S)=f_{k}(S)$. Therefore if $f_{j}(S)+g_{j}(S)$ is an integer
for each $1\leq j\leq h$ and $\chi_{k}=\overline{\chi}_{j}$, then
$g_{j}(S)-g_{k}(S)=\left[f_{j}(S)+g_{j}(S)\right]-\left[f_{k}(S)+g_{k}(S)\right]$
is also an integer for each $1\leq j\leq h$. Hence, the normal mixed Cayley
graph $\text{Cay}(\Gamma,S)$ is Eisenstein integral if and only if
$f_{j}(S)+g_{j}(S)$ is integer for each $1\leq j\leq h$.

######  Lemma 5.1.

The normal mixed Cayley graph $\text{Cay}(\Gamma,S)$ is Eisenstein integral if
and only if $2f_{j}(S)$ and $2g_{j}(S)$ are integer with same parity for each
$1\leq j\leq h$.

###### Proof.

Assume that the normal mixed Cayley graph $\text{Cay}(\Gamma,S)$ is Eisenstein
integral. Then $f_{j}(S)+g_{j}(S)$ and
$g_{j}(S)-g_{k}(S)=\frac{1}{\chi_{j}(1)}\sum\limits_{s\in\overline{S}}\frac{-1}{3}(\sqrt{3}i\chi_{j}(s)-\sqrt{3}i\chi_{j}(s^{-1}))$
are integers for each $1\leq j\leq h$, where $\chi_{k}=\overline{\chi}_{j}$.
By Lemma 4.2,
$\frac{1}{\chi_{j}(1)}\sum\limits_{s\in\overline{S}\cup\overline{S}^{-1}}\chi_{j}(s)\in\mathbb{Z}$.
Since

$2g_{j}(S)=\frac{1}{\chi_{j}(1)}\sum\limits_{s\in\overline{S}\cup\overline{S}^{-1}}\chi_{j}(s)-\frac{1}{\chi_{j}(1)}\sum\limits_{s\in\overline{S}}\frac{i\sqrt{3}}{3}(\chi_{j}(s)-\chi_{j}(s^{-1})),$

we find that $2g_{j}(S)$ is an integer. Therefore,
$2f_{j}(S)=2(f_{j}(S)+g_{j}(S))-2g_{j}(S)$ is also integer with the same
parity with $2g_{j}(S)$.

Conversly, assume that $2f_{j}(S)$ and $2g_{j}(S)$ are integers with the same
parity for each $1\leq j\leq h$. Then $f_{j}(S)+g_{j}(S)$ is an integers for
each $1\leq j\leq h$. Hence the normal mixed Cayley graph
$\text{Cay}(\Gamma,S)$ is Eisenstein integral. ∎

######  Lemma 5.2.

The normal mixed Cayley graph $\text{Cay}(\Gamma,S)$ is Eisenstein integral if
and only if $f_{j}(S)$ and $g_{j}(S)$ are integers for each $1\leq j\leq h$.

###### Proof.

Let $1\leq j\leq h$. According to Lemma 5.1, it is sufficient to prove that
$2f_{j}(S)$ and $2g_{j}(S)$ are integers with the same parity if and only if
$f_{j}(S)$ and $g_{j}(S)$ are integers. If $f_{j}(S)$ and $g_{j}(S)$ are
integers, then clearly $2f_{j}(S)$ and $2g_{j}(S)$ are even integers.
Conversly, assume that $2f_{j}(S)$ and $2g_{j}(S)$ are integers of the same
parity. Since $f_{j}(S)$ is an algebraic integer, then the integrality of
$2f_{j}(S)$ implies that $f_{j}(S)$ is integer. Thus $2f_{j}(S)$ is even
integer, and so by assumption $2g_{j}(S)$ is also even. Hence $g_{j}(S)$ is an
integer. ∎

######  Theorem 5.3.

Let $\text{Cay}(\Gamma,S)$ be a normal mixed Cayley graph. If
$\text{Cay}(\Gamma,S)$ is Eisenstein integral then $\text{Cay}(\Gamma,S)$ is
HS-integral.

###### Proof.

Assume that $\text{Cay}(\Gamma,S)$ is Eisenstein integral. By Lemma 5.2, we
get $f_{j}(S)$ and $g_{j}(S)$ are integers for each $1\leq j\leq h$. It is
important to note that $f_{j}(S)$ is an eigenvalue of the normal simple Cayley
graph $\text{Cay}(\Gamma,S\setminus\overline{S})$. By Theorem 2.6, $f_{j}(S)$
is an integer for each $1\leq j\leq h$ if and only if
$S\setminus\overline{S}\in\mathbb{B}(\Gamma)$. We know
$\frac{1}{\chi_{j}(1)}\sum\limits_{s\in\overline{S}}\frac{-i\sqrt{3}}{3}(\chi_{j}(s)-\chi_{j}(s^{-1}))=g_{j}(S)-g_{k}(S)$
is an integer for each $1\leq j\leq h$, where $\chi_{k}=\overline{\chi}_{j}$.
Using Lemma 4.1, we see that $\overline{S}\in\mathbb{E}(\Gamma)$. Hence by
Theorem 4.4, $\text{Cay}(\Gamma,S)$ is HS-integral. ∎

For $x\in\Gamma$ and $1\leq j\leq h$, define

$C_{x}(j)=\frac{1}{\chi_{j}(1)}\sum_{s\in S^{1}_{x}}\chi_{j}(s)$

where $S^{1}_{x}$ is the smallest symmetric subset of $\Gamma$ containg $x$
that is closed under both conjugate and equivalence relation $\sim$. It is
worthy to note that $C_{x}(j)$ is an eigenvalue of the normal simple(mixed)
Cayley graph $\text{Cay}(\Gamma,S^{1}_{x})$. As a consequence, by Theorem 2.6,
$C_{x}(j)$ is an integer for each $x\in\Gamma$ and $1\leq j\leq h$.

For $y\in\Gamma(3)$, define $S^{2}_{y}$ to be the smallest skew-symmetric
subset of $\Gamma$ containg $y$ that is closed under conjugate as well as
equivalence relation $\approx$. If $\Gamma$ is an abelian group then
$S_{y}^{2}=\langle\\!\langle y\rangle\\!\rangle$ for all $y\in\Gamma(3)$. If
$\Gamma$ is an ambivalent group then $S_{y}^{2}$ does not exist for all
$y\in\Gamma(3)$. Therefore, if $y\in\Gamma(3)$ and $S_{y}^{2}$ exists, then
for $1\leq j\leq h$, define

$T_{y}(j)=\frac{1}{\chi_{j}(1)}\sum_{s\in
S^{2}_{y}}\sqrt{3}i(\chi_{j}(s)-\chi_{j}(s^{-1})).$

Let $j\in\\{1,...,h\\}$. Using $S^{1}_{y}=S^{2}_{y}\cup(S^{2}_{y})^{-1}$, we
get $\frac{C_{y}(j)+T_{y}(j)}{2}$ is an HS-eigenvalue of the normal oriented
Cayley graph $\text{Cay}(\Gamma,S^{2}_{y})$, for each $1\leq j\leq h$.
Therefore, by Theorem 3.2, $\frac{C_{y}(j)+T_{y}(j)}{2}$ is an integer.
Similarly, by Theorem 2.6, $C_{y}(j)$ is also an integer. Which inplies that
$T_{y}(j)$ is an integer.

Let $\Gamma$ is an abelian group or order $n$, $y\in\Gamma(3)$ and
$Irr(\Gamma)=\\{\chi_{1},...,\chi_{n}\\}$. In [16], the auther have proved
that $\frac{T_{y}(j)}{3}$ is integer for each $1\leq j\leq n$.

Consider a alternating group $A_{4}$ and $y=(1,2,3)\in A_{4}(3)$. Then
$S_{y}^{2}=\langle\\!\langle(1,2,3)\rangle\\!\rangle\cup\langle\\!\langle(4,2,1)\rangle\\!\rangle\cup\langle\\!\langle(2,4,3)\rangle\\!\rangle\cup\langle\\!\langle(3,4,1)\rangle\\!\rangle=Cl((1,2,3))$,
and so $T_{y}(1)=T_{y}(4)=0$, $T_{y}(2)=-12$ and $T_{y}(3)=12$. Thus
$\frac{T_{y}(j)}{3}$ is integer for each $1\leq j\leq 4$. Similarly, for
$z=(1,3,2)$ also, $\frac{T_{z}(j)}{3}$ is integer for each $1\leq j\leq 4$.

Let $\Gamma$ be a finite group and $y\in\Gamma(3)$ with $S_{y}^{2}$ exists. We
have seen that $T_{y}(j)$ is an integer for each $1\leq j\leq h$. However, if
$\Gamma$ is an abelian group or $\Gamma=A_{4}$, then $\frac{T_{y}(j)}{3}$ is
an integer for each $1\leq j\leq h$. In the next result, we introduce a
conjecture stating that it is also true in general.

######  Conjecture 5.4.

Let $\Gamma$ be a finite group and $y\in\Gamma(3)$ with $S_{y}^{2}$ exists.
Then $\frac{T_{y}(j)}{3}$ is an integer for each $1\leq j\leq h$.

In the following result, we prove the converse of Theorem 5.3 by combining the
Conjecture 5.4 into a sufficient condition.

######  Theorem 5.5.

Let $\text{Cay}(\Gamma,S)$ be a normal mixed Cayley graph and
$\frac{T_{y}(j)}{3}$ is integer whenever $y\in\overline{S}\subseteq\Gamma(3)$
with $S_{y}^{2}$ exists, for each $1\leq j\leq h$. If $\text{Cay}(\Gamma,S)$
is HS-integral, then $\text{Cay}(\Gamma,S)$ is Eisenstein integral.

###### Proof.

Assume that $\text{Cay}(\Gamma,S)$ is HS-integral and $\frac{T_{y}(j)}{3}$ is
integer whenever $y\in\Gamma(3)$ with $S_{y}^{2}$ exists for each $1\leq j\leq
h$. Then $\text{Cay}(\Gamma,S\setminus\overline{S})$ is integral, and hence
$f_{j}(S)$ is an integer for each $1\leq j\leq h$. By Theorem 4.4, we have
$\overline{S}\in\mathbb{E}(\Gamma)$, and so
$\overline{S}=S_{y_{1}}^{2}\cup...\cup S_{y_{k}}^{2}$, for some
$y_{1},...,y_{k}\in\Gamma(3)$.

$\begin{split}g_{j}(S)&=\frac{1}{2}\sum\limits_{s\in\overline{S}\cup\overline{S}^{-1}}\chi_{j}(s)-\frac{1}{6}\sum\limits_{s\in\overline{S}}i\sqrt{3}\left(\chi_{j}(s)-\chi_{j}(s^{-1})\right)\\\
&=\frac{1}{2}\sum\limits_{l=1}^{k}\sum\limits_{s\in
S_{y_{l}}^{2}\cup(S_{y_{l}}^{2})^{-1}}\chi_{j}(s)-\frac{1}{6}\sum\limits_{l=1}^{k}\sum\limits_{s\in
S_{y_{l}}^{2}}i\sqrt{3}\left(\chi_{j}(s)-\chi_{j}(s^{-1})\right)\\\
&=\frac{1}{2}\sum\limits_{l=1}^{k}C_{y_{l}}(\alpha)-\frac{1}{6}\sum\limits_{l=1}^{k}T_{y_{l}}(\alpha)\\\
&=\frac{1}{2}\sum\limits_{l=1}^{k}\left(C_{y_{l}}(j)-\frac{1}{3}T_{y_{l}}(j)\right).\end{split}$
(12)

Let $1\leq l\leq k$. Since, $\frac{C_{y_{l}}(j)+T_{y_{l}}(j)}{2}$ is an HS-
eigenvalue of the normal oriented Cayley graph
$\text{Cay}(\Gamma,S^{2}_{y_{l}})$, therefore, $C_{y_{l}}(j)$ and
$T_{y_{l}}(j)$ are integers of the same parity. Given that
$\frac{T_{y_{l}}(j)}{3}$ is an integer, and so $C_{y_{l}}(j)$ and
$\frac{T_{y_{l}}(j)}{3}$ are also integers of the same parity for each $1\leq
j\leq h$. We have $C_{y_{l}}(j)-\frac{1}{3}T_{y_{l}}(j)$ is an even integer
for each $1\leq j\leq h$. Then by equation 12, $g_{j}(S)$ is an integer for
each $1\leq j\leq h$. Hence by Lemma 5.2, $\text{Cay}(\Gamma,S)$ is Eisenstein
integral. ∎

Consider the normal mixed Cayley graph $\text{Cay}(A_{4},S)$ of example 4.1.
By Theorem 5.5, $\text{Cay}(A_{4},S)$ is Eisenstein integral. Further, using
Theorem 2.2 and Equation 11, we will caluclate the spectrum of
$\text{Cay}(A_{4},S)$. Let $y=(1,2,3)\in\overline{S}$. Then
$S_{y}^{1}=S^{2}_{y}\cup(S^{2}_{y})^{-1}=\overline{S}\cup\overline{S}^{-1}$,
and so $C_{y}(1)=8,C_{y}(2)=-4,C_{y}(3)=-4,C_{y}(4)=0$. Using equation 12, we
get $g_{1}(S)=4,g_{2}(S)=0,g_{3}(S)=-4$ and $g_{4}(S)=0$. Similarly,
$f_{1}(S)=3,f_{2}(S)=3,f_{3}(S)=3$ and $f_{4}(S)=-1$. Thus by Theorem 2.2 and
Equation 11, the spectrum of $\text{Cay}(A_{4},S)$ is
$\\{[7]^{1},[3+4\omega_{3}]^{1},[-1-4\omega_{3}]^{1},[-1]^{9}\\}$.

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