1. Introduction
The directed strongly regular graph [
1] is one generalization of the undirected strongly regular graphs (SRG), which is an interesting topic in algebraic graph theory.
A
directed strongly regular graph (DSRG) with parameters
is a
k-regular directed graph on
n vertices such that every vertex is on
t 2-cycles (which may be considered as undirected edges), and the number of paths of length two from a vertex
u to a vertex
v is
if there is an arc from
u to
v, and it is
if there is no arc from
u to
v. There is also another definition of a DSRG regarding the adjacency matrix. For a directed graph
X of order
n, its
adjacency matrix is
, where
if
and
are adjacent, and
otherwise. We use
to denote the
identity matrix;
the all-ones matrix. Then
X is a DSRG with parameters
if and only if
and
When
, the DSRG is just the undirected SRG. When
, the DSRG is the doubly regular tournaments [
1]. Therefore we assumed that
in the rest of the paper.
For the SRGs and DSRGs, they share many analog properties. In particular, their eigenvalues are extremely similar. However, for a DSRG, its adjacency matrix is non-symmetric, this leads to more difficulties and makes it an interesting subject. Observe that the DSRGs have several parameters, there has been many constructions oriented to obtain several infinite families of DSRGs, also, some sporadic examples are known in the literature. Although many scholars have studied the existence and constructions of DSRGs for different parameters (one may refer to [
2,
3,
4,
5]), there are also plenty of DSRGs whose existence cannot be determined. As such, the complete characterization of DSRGs is far from being solved.
By using character theory of finite groups, He and Zhang [
6] generalized the semidirect product method in [
2] and obtained a large family of directed strongly regular Cayley graph (DSRCG). Technically, they constructed some DSRCGs over dihedral groups, which partially generalize the earlier results in [
5]. These results reveals that representation theory is a powerful tool in this subject. For more results regarding the interplay between algebraic graph theory and representation theory, one may refer to [
7,
8,
9] and the references therein.
The purpose of this paper is to construct several new infinite families of DSRGs by making use of the representation theory. Borrowing ideas from [
6], we consider the DSRCGs over the metacyclic group
of order
[
10]. Let
be a cyclic multiplicative group of order
n. The metacyclic group
can be viewed as the semidirect product of
of order
n and
of order 4. As mentioned in [
7], if
n is odd, the metacyclic group
is the dicyclic group
. Therefore it would be interesting to consider this group for various applications.
This paper is organized as follows. At first, we give some sufficient and necessary conditions for the Cayley graph with and to be directed strongly regular, and we construct several new classes of DSRCGs over metacyclic groups. Then, for prime p, we characterize the DSRCGs when or .
2. Preliminaries
In this section, we present the fundamental concepts, we also present several lemmas which will be used later. In the sequel, is the n-th cyclotomic field over the rationals, where is the primitive n-th root of unity.
For a multiset M, we define the multiplicity function , where is the number of times the element x appears in M. For two multisets M and N, the sum of M and N is denoted by , then For a positive integer n, the scalar multiplication of M by n is defined as , then we have The difference of M and N is defined as then we have for any For instance, if and , then we have and
For a finite group G with the identity element e (we sometimes use 1 if there is no confusion), and a non-empty subset S of G, we denote by the set . Assume now that , then the graph is called the directed Cayley graph over G with respect to if and (means there is an arc from x to y) if and only if for any
Let
G be a group and
be the complex field. We denote the
group algebra of
G over
by
, and we denote the element of
by
for any multisubset
X of
G. Thus we can write
as
By using the group algebra, we have
Lemma 1 ([
1])
. A Cayley graph of group G with the multiset is a DSRG with parameters if and only if and The following relations will be frequently used in the context.
Lemma 2. For the metacyclic group we have
- (i)
- (ii)
Proof. By relations and the results follow immediately. □
2.1. Fourier Transformation on
The following statement and notations are coincided with [
6,
11]. Let
be the multisubsets of
and
. We denote
,
, and
. The sum of multisubsets
M and
N is
, and the multiplicity function of
is
for any
And let
Then and is a multisubset of
Let
be the multiplicative group of the units in the ring
. Then
has an action on
by multiplication, and hence
is the union of some
-orbits. Each
-orbit consists of all elements of a given order in the additive group
. We denote the
-orbit containing all elements of order
r by
, where
r is a positive divisor of
n. Thus
and
.
We denote all functions
mapping from
to the field
by
By defining the multiplication point-wise, the
-algebra obtained from
will be denoted by
. And the
-algebra obtained from
by defining the multiplication as the
convolution will be denoted by
, where the convolution is defined by:
The
Fourier transformation, as an isomorphsim between the
-algebra
and
, is defined as
For any multisubsets
M and
N of
, we have
Then, for
, where
r is a positive divisor of
n, we have
The following lemmas will be used in the sequel.
Lemma 3 ([
11])
. Let be a function and . Then we have if and only if for some From the Ramanujan’s sums, we have
Let
v be a positive divisor of
n. Then we can define a homomorphism
with
. Then it follows that
Lemma 4 ([
6])
. If is a complex variables function, then we have 2.2. Some Lemmas
Throughout this section, we always assume that p is a prime and is an integer.
Let
be the maximum power of the prime
p that divides
z. Note that the set of divisors of
is
, so all the
-orbits are
, where
We denote by for simplicity hereafter. In particular, .
Note that
is a subgroup of
and
for each
In this section, we assume that
X is a subset of
such that
and
for any
where
m is a positive integer. We also assume
Then we have
Lemma 5 ([
6])
. Let X be a subset of such thatwhere m is a positive integer. Then there exists some integers and satisfy Let and . Then as Thus and .
Lemma 6 ([
6])
. Let and be the sets defined above. Then and form a partition of for some integer Hence Lemma 7 ([
6])
. Let X be a subset of satisfying the condition (5), and , be the sets defined above. If p is an odd prime, then and Lemma 8 ([
6]).
Let X be a subset of satisfies the condition (5), and , be the sets defined above. If , then , and Lemma 9 ([
6]).
Let X be a subset of and be a positive integer. If X satisfies for all , then for some subset of 3. The DSRCGs over
In this section, we will provide several constructions of DSRCGs over .
Let and
We now give a criterion for the Cayley graph to be directed strongly regular.
Lemma 10. The Cayley graph is a DSRCG with parameters if and only if
- (i)
;
- (ii)
;
- (iii)
.
Proof. Thus, from Lemma 1, the Cayley graph
is a DSRCG with parameters
if and only if
Comparing the above two equations, we complete the proof. □
Setting in Lemma 10, we have
Lemma 11. The Cayley graph is a DSRCG with parameters if and only if and Then The following lemma gives a characterization of the Cayley graph to be directed strongly regular by using and .
Lemma 12. The Cayley graph is a DSRCG with parameters if and only if
- (i)
;
- (ii)
;
- (iii)
.
Proof. By Equation (
1), we have
and
From the two equations above and (ii) of Lemma 10, we have
for
. By Equation (
2), we have
and
Then by the two equations above, we have
Using the same method, by (iii) of Lemma 10, we have
for
. Moveover, by applying the Fourier transformation, we have
□
When , we have the following lemma.
Lemma 13. The Cayley graph is a DSRCG with parameters if and only if and Let . Then we have
Lemma 14. The Cayley graph is a DSRCG with parameters then
(1) The function satisfiesfor any (2) There are some integers with satisfywhere (3) If p is an odd prime, then (4) If and , then
Proof. Taking conjugate on Equation (
6), we have
Then the sum of Equations (
6) and (
7) leads to
By Lemmas 6, 7 and 8, we can prove Equations (2), (3) and (4) respectively. □
Next we will present several classes of DSRCGs when or from the above results. In the remainder of this section, v is always assumed to be a positive divisor of n and .
Theorem 1. Let T be a subset of , where v is an odd positive divisor of n, and X be a subset of satisfy the following conditions:
- (i)
- (ii)
Then the Cayley graph is a DSRCG with parameters .
Proof. From (ii), note that . Thus, we have Therefore, by Lemma 11, we get the desired result. □
Example 1. For , we have . Let and Then we have and , where Thus X satisfies the conditions and of the Theorem 1. So we have the Cayley graph is a DSRCG with parameters , where .
Theorem 2. Let T be a subset of , where is an even positive divisor of n. The subset satisfies the following conditions:
- (i)
;
- (ii)
- (iii)
Then the Cayley graph is a DSRCG with parameters .
Proof. By (ii) and (iii), we have and . Therefore, . The result follows from Lemma 11 directly. □
Example 2. For , we have . Let and Then we have . Thus , where , and The set X satisfies the conditions and of Theorem 2. So we have the Cayley graph is a DSRCG with parameters , where .
Theorem 3. Let T be a subset of , where v is an odd positive divisor of n, with . The two subsets satisfy the following conditions:
- (i)
- (ii)
Then the Cayley graph is a DSRCG with parameters .
Proof. Notice that and . Therefore, and Thus the result follows from Lemma 10 directly. □
Example 3. For , we have . Let and Then we have and Thus we have , where Thus the set X satisfies the conditions and of Theorem 3. So we have the Cayley graph is a DSRCG with parameters , where and .
4. Characterization of DSRG
Firstly, we characterize the DSRCGs with .
Theorem 4. Let p be an odd prime and α be a positive integer. Then the Cayley graph is a DSRCG if and only if there is one β with and a subset satisfying the following conditions:
- (i)
- (ii)
Proof. By Theorem 1, we have that satisfies conditions (i) and (ii) is a DSRCG.
Conversely, suppose that the Cayley graph
is a DSRCG with parameters
, where
. From (3) of Lemma 14, we have
for some
proving (ii).
Thus Equation (
6) becomes
By Lemma 9, we have , where T is a subset of , proving (i). □
We now focus on the directed strongly regular Cayley graphs .
Lemma 15. A DSRCG cannot be a Cayley graphs of the form with
Proof. Suppose
By Lemma 14 (2), we have
Similar to the proof of Theorem 4, there is with and a subset such that and . Thus we have that Then , this is impossible. □
Theorem 5. The Cayley graphs is a DSRCG if and only if there exist one β with and a subset satisfying the following conditions:
- (i)
- (ii)
- (iii)
Proof. It follows from Theorem 2 that the Cayley graph satisfying the conditions (i), (ii) and (iii) is a DSRCG.
Conversely, suppose that the Cayley graph
is a DSRCG with parameters
, then we have
by Lemma 15. By Lemma 14 (4) and Equation (
3), we have
for some
Hence
and
Since
and
by Lemma 11, we have
Since
, we have
Thus, by Lemma 11 and Equation (
6), we have
Since
we have
for
Thus, by Lemma 9, we have
where
proving
. Thus we have
So by Equation (
8), we have, for
,
Let
and
. Then
. Thus, by Lemmas 4, 8 and Equation (
9), we have
Since
we have
for
. So
proving
By Lemma 11, we have
then
proving (iii). □
5. Characterization of DSRCG with
Throughout this section, p is assumed to be an odd prime. Let . Then we have
Lemma 16. Suppose the Cayley graph is a DSRCG with . Then if and only if is the union of some -orbits in . Moreover, if is a DSRCG with and , then and , for some .
Proof. Suppose the Cayley graph
is a DSRG with
. If
is a union of some
-orbits in
, then
clearly. If
, by Lemma 12, we have
Thus we have the two eigenvalues are two roots of the quadratic Equation , so we can get . Therefore, by Lemma 3, we have , for some and Thus we have , for some . □
In the following theorem, we characterize certain DSRCG with .
Theorem 6. Let be subsets of with and is a union of some -orbits with . Then the Cayley graph is a DSRCG if and only if the following conditions holds:
- (i)
- (ii)
where and T is a subset of
Proof. By Construction 3, we have that the Cayley graph with conditions (i) and (ii) is a DSRCG.
Conversely, suppose that the Cayley graph is a DSRG with and is a union of some -orbits with . By Lemma 16, we have and , for some . Thus
We claim that
. To prove this claim, we first assume
since this claim holds for
. In fact, if there is an integer
u such that
for some
by Equation (
3), we have
but
a contradiction.
Thus .
The difference of these two equations gives
Since
then we have
Thus we have
Similar to the proof about Case 2 of Theorem 7.2 in [
6], we can get the conditions (i) and (ii). □