Seidel energy for Cayley graphs associated to dihedral group

The studies on the energy of a graph have been actively studied since 1978 and there have been various types of energy of graphs proposed and studied by mathematicians all over the world. One of the many types of energy being studied is Seidel energy, where it is defined as the summation of the absolute values of the eigenvalues of the Seidel matrix of a graph. This research combines the study on three important branches in mathematics, i.e. energy of graph in linear algebra, Cayley graph in graph theory and dihedral group in group theory. The aim of this research is to present some values of Seidel energy of Cayley graphs associated to dihedral groups with respect to certain subsets of the group, namely the subsets of order one and the subsets of order two; {a, a n–1} for n ≥ 3 and {a 2, a n–2} for n ≥ 5. The results are proven by using some properties of special graphs such as complete graph, cycle graph and complete bipartite graph, including some relations between the eigenvalues of a graph, with the Seidel eigenvalues of a graph.


Introduction
The study on Cayley graph was first initiated by Arthur Cayley in 1878 to explain the idea of abstract groups described by a set of generators [1]. It is defined as a graph with the elements of a group G as the vertices and there is an edge joining the vertices g and h in the group G if and only if there is s ∈ S, where S is a subset of G, such that the product of s and g is equal to h. The subset S of G does not include the identity element of the group G and it holds the inverse-closed property where every element of the subset has an inverse under the operation that is also an element of the subset. The Cayley graph of G with respect to the subset S is often denoted as Cay(G; S) [2].
Through the years, the study on Cayley graph has developed and became a significant branch in algebraic graph theory. In 1988, Babai and Seress [3] have presented some results on the diameter of Cayley graphs of symmetric groups including the alternating groups. Not long after, in 1993, Lakshmivarahan et al. [4] have analysed the symmetries in the interconnection networks of a variety of Cayley graphs of permutation groups. The types of symmetry analyzed consist of vertex and edge transitivity, distance regularity and distance transitivity.
In 2000, Friedman [5] has shown in his study that among all sets of n − 1 transpositions which generate the symmetric groups S n , the Cayley graph associated to the set S = Proposition 1. [8] Let S n be symmetric groups. Then Cay(S n , {(ij)}) i, j ∈ {1, 2, . . ., n} are all isomorphic to each other. i.e up to isomorphism, there is exactly one Cayley graph on symmetric groups S n of valency 1.
There have also been several works on the Cayley graphs especially for dihedral groups. In 2006, Wang and Xu [9] worked on the non-normal one-regular and 4-valent Cayley graphs of dihedral groups while in the same year, Kwak and Oh [10] have classified the 4-valent and 6valent one regular normal Cayley graphs on dihedral groups whose vertex stabilizers in Aut(Γ) are cyclic. A k-regular graph is a regular graph with vertices of degree k or also known as a kvalent graph. Meanwhile, vertex stabilizers is an element which fix a vertex of a graph. Besides, Jungreis et al. [11] have studied the hamiltonian of Cayley graphs on groups of low order where a hamiltonian path is a path in an undirected or directed graph that visits each vertex exactly once.
In addition, in 2008, Kwak et al. [12] have explored on one-regular Cayley graphs on dihedral groups of any prescribed valency. Kim et al. [13] have also studied on the Cayley graphs of dihedral groups on the classification of p-valent regular Cayley graphs. They came out with the following theorem and proposition. Theorem 1. [12] Let p be an odd prime number. Then, any p-valent one-regular Cayley graphs on a dihedral groups is isomorphic to one of Cay(D n : l, p) for (n, l, p) ∈ O except at most finitely many ones in which l is positive integers less than n and O be the set of triples (n, l, k) satisfying the following two conditions: (i) For any r, s, t, u(0 ≤ r, s, t, u ≤ k − 1), l[r] + l[s] = l[t] + l[u](modn) if and only if either (r, s) = (t, u) or (r, s) = (u, t).

Proposition 2.
[13] Every prime valent one-regular Cayley graphs on a dihedral group is normal.
Meanwhile, the study on the energy of general simple graphs was inspired from the Hückel Molecular Orbital Theory (HMO) proposed in 1930s. In the early years, the Hückel Molecular Orbital Theory has been used by chemists to approximate the energies related with π−electron orbitals in conjugated hydrocarbon. In 1956, the fact that the method is actually applying a first degree polynomial of the adjacency matrix of a graph was first realized by Günthard and Primas [14]. Later in 1978, the energy of a simple graph has been defined by Gutman [15].
The ordinary energy of a graph Γ is defined as the summation of all positive values of the eigenvalues of the adjacency matrix of the graph. The adjacency matrix A(Γ) of the graph Γ is a square matrix of size n × n, whose ij-entry is equal to 1 if the vertices v i and v j are adjacent and is equal to zero otherwise. The characteristic polynomial of the adjacency matrix, i.e, det(λI n − A(Γ)), where I n is the unit matrix of order n is said to be the characteristic polynomial of the graph Γ and often denoted by f (Γ, x). Since the eigenvalues of a graph Γ are defined as the eigenvalues of its adjacency matrix A(Γ), so they are just the roots of the equation f (Γ, x) = 0, denoted by λ 1 , λ 2 , . . ., λ n .
One of the energy which is closely related by their adjacency matrix is the Seidel energy. The Seidel energy SE(Γ) of a graph Γ is defined as the sum of the absolute values of the eigenvalues of the Seidel matrix of Γ, denoted as SE(Γ) = n i=1 |θ i | where θ i are the eigenvalues of the Seidel matrix of the graph Γ. The Seidel matrix of a simple graph with n vertices and m edges, denoted by S(Γ) = (s ij ) is a real square symmetric matrix of order n defined as s ij = −1 if v i and v j are adjacent, s ij = 1 if v i and v j are not adjacent and 0 if i = j. [16] In order to obtain the Seidel energy of a graph, we need to extract the eigenvalues of the Seidel matrix which we referred as the Seidel eigenvalues. The eigenvalues can be obtained from the roots of the characteristic equation of the matrix. The set of all Seidel eigenvalues is also referred as the Seidel spectrum of a matrix.
In the year 2008, Zhou [17] has presented some findings on the the relations between the main eigenvalues and eigenvectors of an adjacency matrix of a graph which is used in finding the ordinary energy of the graph, with the Seidel matrix of the graph which is used in finding the Seidel energy of the graph. Some of the results emphasized by Zhou in [17] are as given in the following theorem and corollary.

Theorem 2.
[17] Let θ 1 , θ 2 , · · · , θ l be the main eigenvalues of the Seidel matrix S(Γ) and α 1 , α 2 , · · · , α l be the associated orthonormal eigenvectors. Let E be the l × l matrix whose . Then the eigenvalues of M are precisely the main eigenvalues of S(Γ). Furthermore, if b = (b 1 , b 2 , · · · , b l ) T is an eigenvector that corresponding to an eigenvalue μ of M , then l j=1 b j α j is an eigenvector of A(Γ) corresponding to μ. Corollary 1. [17] Let λ 1 , λ 2 , · · · , λ l and θ 1 , θ 2 , · · · , θ l are all main eigenvalues of A(Γ) and S(Γ), respectively. Then Next, in 2012, Haemers [18] has obtained the upper and lower bounds for SE(Γ), characterized the equality for the upper bound, and formulated a conjecture for the lower bound for SE(Γ). Haemers conjectured that the Seidel energy of any graph with n vertices is at least 2n − 2, the Seidel energy of the complete graph with n vertices.
In 2014, Nageswari and Sarasija [19] have presented their findings on the sharp upper and lower bounds for the Seidel energy of connected and disconnected graphs. Their findings are given in the following: The Seidel eigenvalues s 1 , s 2 , · · · , s n of the Seidel matrix of the graph Γ satisfy the following relations: . Theorem 3. [19] If Γ is a connected graph with n vertices and m edges then the inequality [19] If Γ 1 and Γ 2 are two components of a disconnected graph Γ with vertices n 1 and n 2 respectively then the Seidel energy of Γ, SE(Γ) has the following inequality, Corollary 2. [19] If Γ 1 and Γ 2 are two components of a disconnected graph Γ with equal number of vertices say n then the Seidel energy of Γ has the following inequality, In the following year, Ramane et al. [20] have shown that if Γ is regular of order n and of degree r ≥ 3 , then for each k ≥ 2, the Seidel energy of the k-th iterated line graph of Γ depends solely on n and r. Their findings have enable the construction of pairs of non-cospectral, Seidel equienergetic graphs of the same order.

Preliminaries Results
Several forms of the ordinary and Seidel spectrum of special graphs such as complete graph, cycle graph and complete bipartite graph have been found by Brouwer and Haemers [22] as stated in the following propositions. Proposition 4. [22] Consider the undirected n-cycle graph C n . The spectrum of C n consists of the numbers {2cos( 2πj n )}; j = {0, 1, . . ., n − 1}. Proposition 5. [22] Let G be a complete graph K n on n vertices. Its Seidel spectrum is In 2018 and 2019, Fadzil et al. in [23] and [24] have proved the energy of the Cayley graph with respect to the few subsets associated to the dihedral group of order 2n as presented in the following propositions, lemmas and theorems. Proposition 7. [23] Let D 2n be a dihedral group of order 2n where n ≥ 3 and n even. Let S = {a n/2 , b} be a subset of D 2n . The Cayley graphs of D 2n with respect to the set S, Cay(D 2n , {a n/2 , b}), are the cycle graphs n 2 C 4 . Lemma 1. [23] Let D 2n be a dihedral group of order 2n where n ≥ 3 and n even. Let S = {a n/2 , b} be a subset of D 2n . Thus, the spectrum of the Cay(D 2n , {a n/2 , b}), denoted as Spec(Cay(D 2n , {a n/2 , b})) are 0 with multiplicity n and ±2 with multiplicity to n/2. Theorem 5. [23] Let D 2n be a dihedral group of order 2n where n ≥ 3 and S = {a n/2 , b} is a subset of D 2n . The energy of the Cayley graphs of D 2n with respect to the set S, E(Cay(D 2n , {a n/2 , b})) = 2n. Proposition 8. [24] Let D 2n be the dihedral group of order 2n, where n ≥ 3 and X = {b, ab, . . ., a n−1 b} be the generating subset of D 2n . Then, the Cayley graph of D 2n with respect to the generating subset X denoted as Cay(D 2n , {b, ab, . . ., a n−1 b}) is the complete bipartite graph K n,n . Lemma 2. [24] Let D 2n be the dihedral group of order 2n, where n ≥ 3 and X = {b, ab, . . ., a n−1 b} be the generating subset of D 2n . Therefore, the eigenvalues of Cay(D 2n , {b, ab, . . ., a n−1 b}) = K n,n are 0 with multiplicity 2n − 2 and ±n with multiplicity 1.
Recently, in 2020, Akbari et al. [21] have established the validity of Haemers conjecture that the Seidel energy of any graph of order n is at least 2n − 2 and that up to Seidel equivalence, the equality holds for K n .
This paper aims to present the Seidel energy of the Cayley graphs for dihedral group D 2n with respect to the subsets of order up to two. The methodology consists of finding the eigenvalues of the Cayley graphs and thus computing the Seidel energy of the respected Cayley graphs.
This paper is structured as follows: in first section, some introductions and previous studies on the topics are laid out. In second section, some main results on the Seidel energy of the Cayley graphs of the dihedral group are presented in form of lemmas and theorems. Lastly, some conclusion of the results are summarized in the third section.

Main Results
In this section, the Seidel energy of the Cayley graphs associated to the dihedral group D 2n , with respect to subsets of order one and two are presented.

The Seidel Energy of the Cayley Graphs Associated to Dihedral Group With Respect to Subsets of Order One
In the following lemma and theorem, the Seidel eigenvalues and Seidel energy are computed and presented for the Cayley graph associated to the dihedral group with respect to subsets of order one.

Lemma 3.
Let D 2n be the dihedral group of order 2n, where n ≥ 3 and X (1) be a subset of order one of D 2n . Then, the Seidel eigenvalues of the Cayley graphs Cay(D 2n , X (1) ) are ±1 with multiplicity n.
Proof. Suppose D 2n is the dihedral group of order 2n, where n ≥ 3 and X (1) is a subset of order one of D 2n . The Cayley graph of D 2n with respect to subsets of order one, Cay(D 2n , X (1) ) is nK 2 . From Proposition 5, since the Seidel spectrum of a complete graph K n on n vertices is {(1 − n), (1) n−1 }, then the Seidel eigenvalues of Cay(D 2n , X (1) ) = nK 2 are θ i = ±1 with multiplicity n.
The Seidel energy of the Cayley graph associated to the dihedral group of order 2n with a subset of order one is found and presented in the following theorem.

Theorem 7.
Let D 2n be the dihedral group of order 2n, where n ≥ 3 and X (1) be a subset of order one of D 2n . Then, the Seidel energy of the Cayley graphs Cay(D 2n , X (1) ) is 2n.
An example is provided to illustrate the previous theorem.

The Seidel Energy of the Cayley Graphs Associated to Dihedral Group With Respect to Subsets of Order Two
This subsection presented the results on the Seidel energy of the Cayley graph associated to the dihedral group of order 2n with respect to the subsets of order two in the form of theorems.

Theorem 8.
Let D 2n be the dihedral group of order 2n, where n ≥ 3 and X (2) = {a, a n−1 } be a subset of order two of D 2n . Then, the Cayley graphs of D 2n with respect to the subset X (2) , denoted by Cay(D 2n , {a, a n−1 }) is two cycle graphs of length n, denoted as 2C n . Proof. Suppose D 2n is the dihedral group of order 2n, where n ≥ 3 and X (2) = {a, a n−1 } is a subset of order two of D 2n . Then, by using the definition of Cayley graph, it is shown that the followings are two cycles of length n for Cay(D 2n , {a, a n−1 }). (2) and (a n−1 )(1) −1 = a n−1 ∈ X (2) , then a i is adjacent to a i+1 for 0 ≤ i ≤ n − 2 and a n−1 is adjacent to 1. Thus, it is proven that (1) is a cycle.
Next (2) and (a n−1 b)(b) −1 = a n−1 bb −1 = a n−1 ∈ X (2) , then a i b is adjacent to a i+1 b for 0 ≤ i ≤ n − 2 and a n−1 b is adjacent to b. Thus, it is proven that (2) is a cycle.
Since all Cayley graphs of D 2n with respect to the subset X (2) = {a, a n−1 } are 2-regular graphs, then there is no edge between both cycles. Therefore, Cay(D 2n , {a, a n−1 }) is two cycle graphs of length n, denoted as 2C n . Theorem 9. Let D 2n be the dihedral group of order 2n, where n ≥ 3 and X (2) = {a, a n−1 } be a subset of order two of D 2n . Then, the Seidel energy of the Cayley graphs Cay(D 2n , {a, a n−1 }) is the dihedral group of order 2n, where n ≥ 3 and X (2) = {a, a n−1 } is a subset of valency two of D 2n . By Theorem 8, the Cayley graph of D 2n with respect to the subset {a, a n−1 }, denoted as Cay(D 2n , {a, a n−1 }) is 2C n . In order to find the Seidel energy of the Cayley graph, Corollary 1 is used. Obviously, from Proposition 4, the eigenvalues of Cay(D 2n , {a, a n−1 }) are λ i = {2 cos( 2πj n ); j = 0, 1, . . ., n − 1} with multiplicity 2. Since Cay(D 2n , {a, a n−1 }) is a 2-regular graph, its Seidel spectrum is determined by its adjacency spectrum as given in the following: l i=1 (2λ i + θ i ) = n − l where λ i and θ i are the eigenvalues of l × l adjacency matrix A(Γ) and l × l Seidel matrix A * (Γ) respectively. Therefore, by rearranging the above equation gives the following equation: Thus, the Seidel energy of Cay(D 2n , {a, a n−1 }) is the summation of its Seidel eigenvalues as given in the following: Therefore, the Seidel energy of the Cayley graphs Cay(D 2n , {a, a n−1 }) is as given in the following: |8 cos( 2πi n )|.
Next theorem presents the Seidel energy for the Cayley graph of the dihedral group with respect to the subset {a 2 , a n−2 } for the case n is odd. Theorem 10. Let D 2n be the dihedral group of order 2n, where n ≥ 5 and n is odd. Let X (2) = {a 2 , a n−2 } be a subset of order two of D 2n . Then, the Cayley graphs of D 2n with respect to the subset X (2) , denoted by Cay(D 2n , {a 2 , a n−2 }) is two cycle graphs of length n, denoted as 2C n .
Proof. Suppose D 2n is the dihedral group of order 2n, where n ≥ 5 and n is odd. Let X (2) = {a 2 , a n−2 } be a subset of order two of D 2n . Then, by using the definition of Cayley graph, it is shown that the followings are two cycles of length n for Cay(D 2n , {a 2 , a n−2 }).
Next, it is proven that (4) is a cycle based on the followings.
Therefore, both (3) and (4) are cycles. Since all Cayley graphs of D 2n with respect to the subset X (2) are 2-regular graphs, then there is no edge between both cycles. Thus, the Cayley graphs of D 2n with respect to the subset X (2) = {a 2 , a n−2 }, where n ≥ 5 and n is odd, is the two cycle graphs of length n, denoted as 2C n . Theorem 11. Let D 2n be the dihedral group of order 2n, where n ≥ 3 and n is odd. Let X (2) = {a 2 , a n−2 } be a subset of valency two of D 2n . Then, the Seidel energy of the Cayley graphs Cay(D 2n , {a 2 , a n−2 }) is n i=1 |8 cos( 2πi n )|. Proof. Suppose D 2n is the dihedral group of order 2n, where n ≥ 3 and n is odd. Let X (2) = {a 2 , a n−2 } be a subset of valency two of D 2n . By Theorem 10, the Cayley graph of D 2n with respect to the subset {a 2 , a n−2 }, denoted as Cay(D 2n , {a 2 , a n−2 }) is 2C n . Its easy to see that the eigenvalues of Cay(D 2n , {a 2 , a n−2 }) are λ i = {2 cos( 2πj n ); j = 0, 1, . . ., n − 1} with multiplicity 2. Similarly, as in the proof of Theorem 9, the Seidel energy of the Cayley graphs Cay(D 2n , {a 2 , a n−2 }) where n ≥ 3 and n is odd is as given in the following: An example is provided to illustrate the previous theorem. Let D 10 be the dihedral group of order 10, where D 10 = a, b|a 5 = b 2 = 1, bab = a −1 and X (2) = {a 2 , a 3 } be the subset of D 10 of valency two. Then, clearly, the Cayley graph Cay(D 10 , {a 2 , a 3 }) is the union of two cycle graphs of five vertices, 2C 5 . Then, the Seidel energy of Cay(D 10 , {a 2 , a 3 }) is given as 5 i=1 |8 cos( 2πi 5 )|. The following theorem presents the Seidel energy for the Cayley graph of the dihedral group with respect to the subset {a 2 , a n−2 } for the case n is even.
Theorem 12. Let D 2n be the dihedral group of order 2n, where n ≥ 5 and n is even. Let X (2) = {a 2 , a n−2 } be a subset of order two of D 2n . Then, the Cayley graphs of D 2n with respect to the subset X (2) , denoted by Cay(D 2n , {a 2 , a n−2 }) is four cycle graphs of length n 2 , denoted as 4C n 2 . Proof. Suppose D 2n is the dihedral group of order 2n, where n ≥ 5 and n is even. Let X (2) = {a 2 , a n−2 } be a subset of order two of D 2n . Assume that n = 2k, then, by using the definition of Cayley graph, it is proven that the followings are four cycles of length k for Cay(D 2n , {a 2 , a n−2 }).
Therefore, all of them are cycles. There is no edge between all cycles since all Cayley graphs of D 2n with respect to the subset X (2) are 2-regular graphs. Thus, the Cayley graphs of D 2n with respect to the subset X (2) = {a 2 , a n−2 }, where n ≥ 5 and n is even, is the four cycle graphs of length n 2 , denoted as 4C n 2 .