Eigenvalues of Antiadjacency Matrix of Cayley Graph of Z n

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Introduction
Cayley graph has always been an interesting subject in mathematics, since it allows us to understand group theory by using graph theory and vice versa. Let G be a group with 0 G as its identity element and S be an inverse-closed (that is S = S −1 , where S −1 = {s −1 : s ∈ S}) subset of G − {0 G }. Cayley graph on group G with connection set S, denoted by Cay(G, S) is defined as graph with G as its vertex set and arcs (x, y) for every pair (x, y) which satisfies xy −1 ∈ S [9]. One of the most active research area in Cayley graph is considering the eigenvalues of the adjacency matrix of Cayley graphs.
There are many intriguing research problems related to eigenvalues research of the adjacency matrix, such as the problems of finding integral Cayley graphs. Integral graph itself is defined as graphs whose eigenvalues of adjacency matrix are all integer. The search for integral graphs began www.ijc.or.id from the research done by Harary and Schwenk [5] in 1974. There are many research about Cayley integral graphs on various groups, such as permutation group ( [4], [6]), dihedral group ( [1], [10]), and Z n group ( [7], [14]).
In his research, So [14] has found a necessary and sufficient condition for Cayley graph of Z n to be integral, which is stated in the following theorem Theorem 1.1. [14] Let n be an integer greater than 1 and d be a positive factor of n. Define S n (d) = {a : 0 < a ≤ n, gcd(a, n) = d}. Then Cay(Z n , S) is an integral graph if and only if S is the union of some S n (d).
Klotz and Sander [7] in 2007 also found the sufficient condition in Theorem 1.1 through different method. They introduced the concept of gcd graph, which is graph with vertex set Z n and arcs (x, y) if and only if gcd (x − y, n) is a divisor of n. He then proved that gcd graphs are all integral graphs. It can be seen that the graphs mentioned in [7] are equivalent with the graphs mentioned in [14]. In 2017, Mirafzal et. al. [11] proved that Cay(Z 2n , (Z 2n − {0}) − {n}) is an integral graph for all integers n ≥ 2. Moreover, they found that all the eigenvalues of the adjacency matrix of the graph are 2n − 2, 0, and −2 and their multiplicities are 1, n, n − 1, respectively.
There are many matrix representations of graphs, such as antiadjacency matrix. Antiadjacency matrix of graph G, usually denoted by B, is a matrix B = J − A with J equals to n × n matrix with all of its entries are 1 and A is the adjacency matrix of graph G [3]. Stin et al. [15] studied about the eigenvalues of the antiadjacency matrix of cyclic directed prism graph. Murni et al. [12] also researched about the antiadjacency matrix of inverse graph of Z n . They found several spectra of inverse graph of Z n for few n. Oktradifa et al. [13] investigated the eigenvalues of antiadjacency matrix of directed unicyclic helm graph.
There are not many research on the eigenvalues of the antiadjacency matrix of Cayley graphs of groups. Thus, in this research, we are interested to explore the properties of the eigenvalues of the antiadjacency matrix of Cayley graphs.

Preliminary Results
Matrix C is said to be a circulant matrix if C is a square matrix of size n × n which can be represented as follows where ξ = exp(2πi/n) and t = 0, 1, 2, ..., n − 1.
We make the following observation now, which we will apply frequently in the next section.
Proof. For t = 0 mod n, we can write t as kn with k ∈ Z and so For other values of t, t = kn + c, with k an integer and c ∈ {1, ..., n − 1}. Observe that ξ kn+c = ξ c and so n−1 j=0 ξ tj is a geometric series with ratio ξ c , so

Main Results
In this section, we present our results about the eigenvalues of the antiadjacency matrix of Cay(Z n , S). Not only that, we also present the relation between eigenvalues of the antiadjacency matrix of Cay(Z n , S) and eigenvalues of two other matrix representations of Cayley graph of Z n , which are the antiadjacency matrix of Cay(Z n , (Z n − {0}) − S) and the adjacency matrix of Cay(Z n , S).
Before we go into the results, we introduce the following simple lemma, which is very essential for our proof in the next two theorems. Proof. By Lemma 2.1, the adjacency matrix A of Cay(Z n , S) is a circulant matrix. Hence, matrix A can be written in the following form .. a n−1 a n−1 a 0 a 1 ... a n−2 a n−2 a n−1 a 0 ... a n−3 . . .
By the definition of antiadjacency matrix, we have B can be written as Conclusively, B is a circulant matrix. Now, with the aid of Lemma 3.1, we shall present our first theorem which gives the connection between the eigenvalues of the adjacency matrix of Cay(Z n , S) and the eigenvalues of the antiadjacency matrix of Cay(Z n , S).
Theorem 3.1. If the spectrum of the adjacency matrix A of Cay(Z n , S) is then the spectrum of the antiadjacency matrix B of Cay(Z n , S) is Proof. From Corollary 2.1, matrix A is a circulant matrix, so it can be written as .. a n−1 a n−1 a 0 a 1 ... a n−2 a n−2 a n−1 a 0 ... a a j ξ tj , with ξ = exp( 2πi n ) and t = 0, 1, 2, .., n − 1. From the definition of antiadjacency matrix B of Cay(Z n , S), matrix B can be written as Since matrix B is circulant by Corollary 3.1, then by Theorem 2.1, the eigenvalues of B are with ξ = exp( 2πi n ) and t = 0, 1, 2, . . . , n − 1. For t = 0, by using Lemma 2.1, Equation 2 becomes On the other hand, for nonzero t, by Lemma 2.1, Equation 2 becomes Hence, we obtain the following: for t = 1, 2, ..., n − 1, n − λ t , for t = 0.
Therefore, an eigenvalue λ 0 of matrix A corresponds to an eigenvalue n − λ 0 of matrix B. For other n − 1 eigenvalues of matrix A, λ i , B has −λ i as its eigenvalues. Thus,

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Eigenvalues of Antiadjacency Matrix of Cayley Graph of Z n | Juan Daniel et al.
We present our another main result, which describes the relation between the eigenvalues of the antiadjacency matrix of Cay(Z n , S) and Cay(Z n , (Z n − {0}) − S).    Thus,
From Lemma 3.1, B is also a circulant matrix. By Theorem 2.1, the eigenvalues of B are a j ξ tj for t = 0, 1, 2, . . . , n − 1 and ξ = exp 2πi n . Then, Since a 0 is always zero, we have Now, we divide the proof in two cases for t = 0 and t = 0. For t = 0, from Equation 3 and Lemma 2.1, we have For t = 0, by Equation 3 and Lemma 2.1, we have Conclusively, for t = 1, 2, ..., n − 1, n + 1 − λ t , for t = 0.
Therefore, an eigenvalue λ 0 of matrix B corresponds to an eigenvalue n + 1 − λ 0 of matrix B . For n − 1 other eigenvalues λ i of B, B has 1 − λ i as its eigenvalue, so we obtain

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Eigenvalues of Antiadjacency Matrix of Cayley Graph of Z n | Juan Daniel et al.
By using Theorem 1.1 and Theorem 3.1, we can easily give the necessary and sufficient condition such that the eigenvalues of the antiadjacency matrix of Cay(Z n , S) are integer. Corollary 3.1. All the eigenvalues of the antiadjacency matrix of Cay(Z n , S) are integers if and only if S is the union of some S n (d).
As another application of Theorem 3.1, we give the following result regarding the eigenvalues of the antiadjacency matrix of Cay(Z n , S) for the connection set S ⊆ Z n − {0} such that S ∪ {0} is a subgroup of Z n . Corollary 3.2. Let S be a subset of Z n − {0} such that S ∪ {0} is a subgroup of (Z n , + modn ). The spectrum of antiadjacency matrix B of Cay(Z n , S) is Since n = km and |S| = k − 1, then the spectrum above can be written as We close this section by giving an example of how to generally find the eigenvalues of the antiadjacency matrix of Cay(Z n , S).
Example 3.1. Let n be a positive integer greater than 1. We illustrate how Theorem 3.1 is applied to find the eigenvalues of the antiadjacency matrix of Cayley graphs of Z n . In this example, we only demonstrate it for the case of Cay(Z 2n , {1, n, 2n − 1}). Refer to Figure 1 for the case n = 4.

Conclusion
In this paper, we examined the relation between eigenvalues of the antiadjacency matrix of Cay(Z n , S) and Cay(Z n , (Z n − {0}) − S). We also examined the relation between eigenvalues of the antiadjacency matrix of Cay(Z n , S) and eigenvalues of the adjacency matrix of Cay(Z n , S). Those results were mainly obtained by using the property of circulant matrix. We also obtained that all of the eigenvalues of the antiadjacency matrix of Cay(Z n , S) are integer if and only if S is the union of some S n (d) = {a : 0 < a ≤ n, gcd(a, n) = d}.
Open Problem: What are the sufficient and necessary conditions for all the eigenvalues of the antiadjacency matrix of Cayley graphs of other groups, such as dihedral group and permutation group, to be an integer?