Seidel Laplacian and Seidel Signless Laplacian Spectrum of the Zero-divisor Graph on the Ring of Integers Modulo n

Let G be a simple graph of order n and let S p G q be the Seidel matrix of G , deﬁned as S p G q (cid:16) r s ij s where s ij (cid:16) (cid:1) 1 if the vertices v i and v j are adjacent and s ij (cid:16) 1 if the vertices v i and v j are not adjacent and s ij (cid:16) 0 if i (cid:16) j . Let D S p G q (cid:16) diag p n (cid:1) 2 d 1 (cid:1) 1 , n (cid:1) 2 d 2 (cid:1) 1 , ..., n (cid:1) 2 d n (cid:1) 1 q be the diagonal matrix where d i denotes the degree of the i th vertex of G . The Seidel Laplacian matrix of a graph G is deﬁned as SL p G q (cid:16) D S p G q (cid:1) S p G q and the Seidel signless Laplacian matrix of a graph G is deﬁned as SL (cid:0) p G q (cid:16) D S p G q (cid:0) S p G q . The zero-divisor graph of a commutative ring R , denoted by Γ p R q , is a simple undirected graph with all non-zero zero-divisors as vertices and two distinct vertices x, y are adjacent if and only if xy (cid:16) 0 . In this paper, we ﬁnd the Seidel polynomial and Seidel Laplacian polynomial of the join of two regular graphs using the concept of schur complement and coronal of a square matrix. Also we describe the computation of the Seidel Laplacian and Seidel signless Laplacian eigenvalues of the join of more than two regular graphs, using the well known Fiedler’s lemma and apply these results to describe these eigenvalues for the zero-divisor graph on Z n . Further we ﬁnd the Seidel Laplacian and Seidel signless Laplacian spectrum of the zero-divisor graph of Z n for some values of n , say n (cid:16) p 3 , p 4 , pq, p 2 q , where p, q are distinct primes. We also prove that 0 is a simple Seidel Laplacian eigenvalue of Γ p Z n q , for any n .


Introduction
Throughout this paper, G denotes a simple, finite, undirected and connected graph. If G has n vertices, then the adjacency matrix, ApGq ra ij s n¢n where, a ij 1 if v i v j and a ij 0 otherwise. Van Lint and Seidel [1] introduced the Seidel matrix of G, defined as SpGq rs ij s where, s ij if v i & v j 0; otherwise . Clearly, SpGq J ¡ I ¡ 2ApGq. Also, if G denotes the complement of a graph G, then SpGq ¡SpGq. The Seidel spectrum of G is denoted by spec S pGq. Let D S pGq diagpn ¡ 2d 1 ¡ 1, n ¡ 2d 2 ¡ 1, ..., n ¡ 2d n ¡ 1q be the diagonal matrix where d i is the degree of the vertex v i . The Seidel Laplacian matrix [2] of a graph G is defined as SLpGq D S pGq ¡ SpGq 918 Seidel Laplacian and Seidel Signless Laplacian Spectrum of the Zero-divisor Graph on the Ring of Integers Modulo n and the Seidel signless Laplacian matrix of a graph G is defined as For a complete graph K n , SLpK n q J n ¡ nI n and SL pK n q p2 ¡ nqI n ¡ J n .
For a null graph, K n , SLpK n q nI n ¡ J n and SL pK n q pn ¡ 2qI n J n .

Basic definitions and notations
Let ΦpG; xq and SpecpGq denote the characteristic polynomial and the spectrum of G respectively. Φ SL pG; xq and Φ SL pG; xq denote the Seidel Laplacian polynomial and Seidel signless Laplacian polynomial of a graph G. In this paper, 1 n denotes the all-one column vector of order n ¢ 1, and φpnq is the number of positive integer less than n and relatively prime to n.
Definition 2.1. The join of two graphs G 1 and G 2 ; denoted by G 1 G 2 , is obtained by joining each vertex of G 1 to all the vertices of G 2 . The sum of the entries of the matrix pxI ¡ Aq ¡1 is defined as the coronal of A, and is denoted by Γ A pxq. It can be seen that, Γ A pxq p1 n q T .pxI ¡ Aq ¡1 .1 n . We note that, if k is the sum of each row, then Γ A pxq n x ¡ k . I. Beck [5] initiated the idea of zero-divisor graph GpRq of a commutative ring R in connection with some coloring problems and in 1999, Anderson and Livingston [6], modified the definition of zero-divisor graph as a simple graph Γ pRq where, only nonzero zero-divisors of R are considered as vertices. The concept of compressed zero-divisor graph; determined by the equivalence classes of the zero-divisors of R, was introduced by Mulay [7], with the purpose of simplifying the representation of Γ pRq In [8,9], the authors describe the adjacency matrix, eigenvalues and some graph parameters of the zero-divisor graphs Γ pZ p 2 q q, Γ pZ p 2 q 2 q and Γ pZ p k q. S. Chattopadhyay et.al [10], have explored the combinatorial structure of Γ pZ n q as the generalized union of its induced subgraphs. The combinatorial properties of the proper divisor graph on n have been investigated in [11]. It is very interesting and challenging that the combinatorial and spectral properties of zero-divisor graphs can be studied in terms of its compressed graph.

Fiedler's Lemma and its generalisation
The Abel's impossibility theorem implies that the algebraic determination of all the zeroes of a polynomial of degree greater than or equal to five, is impossible in closed form. So, we need resort to the tools of Linear Algebra to explore the eigenvalues of large matrices by means of its sub matrices. In this direction, the famous Fiedler's lemma is very relevant to find the eigenvalues of large symmetric matrices. In [12], H.S.Ramane et.al find the Seidel Laplacian polynomial of the join of two regular graphs. In this section, we find the same in a fairly shorter method, by applying the following lemma and extend the result to the join of more than two regular graphs.
If G is a kregular graph of order n, and k, λ 2 , ..., λ n are the adjacency eigenvalues of G, then .., θ n ¡1 ¡ 2λ n are the Seidel eigenvalues of G.
Theorem 3.1. Let G i be r i regular of order n i , for i 1, 2. Then the Seidel Laplacian polynomial of G 1 G 2 is given by Φ SL pG 1 G 2 ; xq xpx n 1 n 2 q px n 1 qpx n 2 q ¤ Φ SL pG 1 ; x n 2 q ¤ Φ SL pG 2 ; x n 1 q. Proof. Let SpG 1 q and SpG 2 q denote the Seidel adjacency matrices of G 1 and G 2 respectively. Then, the Seidel Laplacian matrices of G 1 and G 2 are given as follows.
The authors D.M. Cardoso et al [ 15] and Magi.P.M. et al [16] have used the above result to find distance related spectrum of the generalized join of graphs.

Seidel Laplacian spectrum of the joined union of regular graphs
Consider G rH 1 , H 2 , ..., H k s, where the vertices of G are labeled as 1, 2, ..., k, and the Seidel matrix SpGq rs ij s. Let H j be r j -regular and |V pH j q| n j , for every j 1, 2, ..., k. Let SpH j q and SLpH j q denote the Seidel matrix and Seidel Laplacian matrix of H j , j 1, 2, ..., k. The degree of each vertex of H j , in the joined graph G rH 1 , where τ j °k i1 s ij n i .
Thus, we see that, the Seidel Laplacian matrix of the Gunion of H 1 , H 2 , ..., H k is given by where τ j °k i1 s ij n i . Remark 4.1. Since for a k-regular graph G of order n, SLpGq pn ¡ 2k ¡ 1qI n ¡ SpGq, from Lemma 3.2 it follows that, SLpGq has an eigenvalue 0 with multiplicity at least 1. For example, the Seidel Laplacian spectrum of the cycle C 4 , which is 2- . However, the Seidel Laplacian matrix of complete graphs and null graphs (complement of complete graphs) has an eigenvalue 0 with multiplicity 1.
.., H k s and S rs ij s k¢k is the Seidel matrix of G and H j is r j -regular and |V pH j q| n j , for every j 1, 2, ..., k. Let tσ SL j1 0, σ SL j2 , ..., σ SL jnj u be the Seidel Laplacian eigenvalues of H j , for j 1, 2, ..., k. Then, where τ j °k i1 s ij n i and Proof. Since H j is regular, 0 is a Seidel Laplacian eigenvalue of H j with eigenvector 1 nj , for every j. Thus from (5), it is evident that each of the diagonal blocks SLpH j q τ j I nj is a symmetric matrix which has τ j as an eigenvalue with eigenvector 1 nj , j 1, 2, ..., k.

As in (3), taking
M j SLpH j q τ j I nj , pα ij ,j , u ij ,j q pτ j , 1 c n j 1 nj q and the real numbers ρ l,q ¡s lq c n l n q for l t1, 2, ..., k ¡ 1u, q tl 1, ..., ku, and applying Theorem 3.2, we find that, where τ j °k i1 s ij n i and r S " " " !
Obviously, ρ l,q ¡s lq c n l n q 5 c n l n q if lq E(G) ¡ c n l n q if lq E(G) for l t1, 2, ..., k ¡ 1u, q tl 1, ..., ku. Assign the weight n j |V pH j q| to the vertex j of G for j 1, 2, ..., k and consider the matrix W which is a diagonal matrix of vertex weights, The vertex weighted combinatorial Laplacian matrix of graphs can be seen in [17]. It is easy to verify that T SL pGq W ¡ 1 2 r SW 1 2 . Thus r S and T SL pGq are similar and hence specp r Sq specpT SL pGqq. Hence from (6), 5 The zero-divisor graph Γ pZ n q The commutative ring Z p is an integral domain for any prime p and so it is quite trivial to study its zero-divisor graph. Hence in the following sessions, it is assumed that n is not a prime. It is very relevant to note that Γ pZ p 2 q is a complete graph on p ¡ 1 vertices. Also Γ pZ 8 q, Γ pZ pq q are complete bipartite graphs. Using elementary number theory, the number of non zero zero-divisors is calculated to be n ¡ φpnq ¡ 1 [9]. For distinct primes p 1 , p 2 , ..., p r , let n p n1 1 ¤ p n2 2 ¤ ¤ ¤ p nr r be the canonical decomposition of n. A proper divisor of n is a positive divisor d such that d # n, 1 d n. Let ppnq denote the number of proper divisors of n. Then, ppnq σ 0 pnq ¡ 2, where σ k pnq is the sum of k powers of all divisors of n, including n and 1.
Let Spdq tk Z n : gcdpk, nq du. Then tSpd 1 q, Spd 2 q, ..., Spd ppnq qu forms an equitable partition of V pΓpZ n qq Also, |Spd i q| φp n d i q, for every i 1, 2, ...ppnq. The following Lemmas are inevitable in the analysis of Γ pZ n q.
The proper divisor graph, denoted by Υ n plays a vital role in the analysis of Γ pZ n q, the vertices of which are labeled as d 1 , d 2 , ..., d ppnq . See [11]. Also d i d j in Υ n if and only if n divides the product d i d j .
Proof. In the proper divisor graph Υ n , the vertices d i and d j are adjacent if and only if n # d i d j . Thus if S rs ij s ppnq¢ppnq denotes the Seidel adjacency matrix of Υ n , then By Lemma 5.2, Γ pZ n q is the Υ n -join of ΓpSpd 1 qq, ΓpSpd 2 qq, ..., ΓpSpd ppnq qq. The induced subgraphs ΓpSpd j qq are either a complete graph or a null graph on φp n dj q vertices.
By Lemma 5.1, The Seidel Laplacian matrix of K n and K n are given by SLpK n q J n ¡ nI n , SLpK n q nI n ¡ J n . Thus, if S j denotes the j th diagonal block in the Seidel Laplacian matrix of Γ pZ n q, which corresponds to the vertices of ΓpSpd j qq, then, S j SLpΓpSpd j qqq τ j I φp n d j q where, τ j °p pnq i1 s ij φp n di q, from (5). Hence, Thus, the result follows from (5), taking G Υ n and H j ΓpSpd j qq. Applying Theorem 4.1 and Theorem 5.1, we determine the Seidel Laplacian spectrum of Γ pZ n q in the following corollary.
where τ j °p pnq i1 s ij φp n di q, j 1, 2, ..., ppnq. Definition 5.1. [18] A matrix ra ij s is strictly diagonally dominant if in every row of the matrix, the magnitude of the diagonal entry is strictly greater than the sum of the magnitudes of all other non-diagonal entries, that is if, |a ii | ¡°j $i |a ij |, for all i. Proof. First, we show that 0 is an eigenvalue of the matrix T SL pGq of multiplicity 1. For this, we prove that T SL pGq is singular with nullity 1. Arrange the proper divisors of n in the ascending order, d 1 d 2 ... d ppnq . It is obvious that, φp n d1 q ¡ φp n di q, i 2, 3, ..., ppnq. We note that T SL pGq is a square matrix of size ppnq. Since, τ j °p pnq Hence it follows from (9), that the sum of each row of T SL pGq is zero. The first column of T SL pGq can be transformed to the zero column on adding 2 nd , 3 rd , ..., ppnq th columns to it. Hence T SL pGq is singular which implies that 0 is an eigenvalue of T SL pGq. To prove that the multiplicity is 1, it suffices to prove that the rank of T SL pGq is ppnq ¡ 1. For this, consider the matrix T obtained from T SL pGq by deleting the first row and first column of T SL pGq. Since Thus, T is strictly diagonally dominant. For example, consider the first row of T , say rτ 2 , ¡s 2,3 φp n d3 q, ¡s 2,4 φp n d4 q, ..., ¡s 2,ppnq φp n d ppnq qs. Since, τ 2 s 1,2 φp n d1 q s 2,3 φp n d3 q .... s 2,ppnq φp n d ppnq q, it follows that, |τ 2 | ¡ |s 2,3 φp n d3 q| ... |s 2,ppnq φp n d ppnq q|. Hence by Theorem 5.2, T is non-singular, and hence the rank of T SL pGq is ppnq ¡ 1.
By Corollary 5.1, the remaining Seidel eigenvalues of Γ pZ n q are φp n dj q °p pnq i1 s ij φp n di q and°p pnq i1 s ij φp n di q ¡ φp n dj q, neither of which is zero. This proves the theorem.
Theorem 5.4. For distinct primes p and q, p q, the Seidel Laplacian spectrum of Γ pZ pq q is given by Proof. The proper divisors of pq are p and q. By Lemma 5.1 and Lemma 5.2, the zero-divisor graph Γ pZ pq q is the join of ΓpSppqq and ΓpSpqqq, where ΓpSppqq K q¡1 and ΓpSpqqq K p¡1 . That is Γ pZ pq q K q¡1 K p¡1 .
Clearly spec SL pK q¡1 q 4 0 q ¡ 1 1 q ¡ 2 B and spec SL pK p¡1 q 4 . And the result follows from Theorem 3.1.
Theorem 5.5. For any prime p, the Seidel Laplacian spectrum of Γ pZ p 3 q is Proof. The proper divisors of p 3 are p and p 2 . By lemma 5.1 it can be seen that the subgraphs of Γ pZ p 3 q, induced by Sppq and Spp 2 q are K ppp¡1q and K p¡1 respectively. Also by Lemma 5.2, 924 Seidel Laplacian and Seidel Signless Laplacian Spectrum of the Zero-divisor Graph on the Ring of Integers Modulo n spec SL pK ppp¡1q q Hence, the result follows from Theorem 3.1.
Theorem 5.6. For any prime p, the Seidel Laplacian spectrum of the zero-divisor graph Γ pZ p 4 q is Proof. The divisors of p 4 are p, p 2 , p 3 . The proper divisor graph of p 4 is the path P 3 , in which p p 3 p 2 . The subgraph induced by Sppq is the null graph K p 2 pp¡1q , whereas the subgraphs induced by Spp 2 q and Spp 3 q are the complete graphs K ppp¡1q and K p¡1 respectively. It can be seen that, Γ pZ p 4 q P 3 rK p 2 pp¡1q , K ppp¡1q , K p¡1 s. Applying Corollary 5.1, we see that, p 3 ¡2p 1, p 3 ¡2p 2 1, 1¡p 3 are Seidel Laplacian eigenvalues of Γ pZ p 4 q with multiplicities p 3 ¡p 2 ¡1, p 2 ¡p¡1 and p¡2 respectively. And the remaining three Seidel Laplacian eigenvalues of Γ pZ p 4 q are the eigenvalues of the matrix, It can be seen that, the above matrix has three eigen values, λ 1 0, λ 2 1 ¡ p 3 , λ 3 p 3 ¡ 2p 1.

Seidel signless Laplacian spectrum of the join of regular graphs
We note that, each diagonal block in the Seidel signless Laplacian matrix of the join of two regular graphs is a symmetric matrix which bears an eigenvalue with all-one vector as the corresponding eigenvector, which facilitates the use of Fiedler's Lemma in the investigation of its spectrum. Since the main theorems of this section are in the same frame work of Fiedlers Lemma, we avoid repetition in proofs, except Theorem 6.1, where the concept of Schur complement and Coronal of a square matrix are incorporated.
where α is a real number.
As in section:4, Theorem 3.2 can be applied to find the Seidel signless Laplacian spectrum of the join of regular graphs.
Consider G rH 1 , H 2 , ..., H k s, where G is a simple connected graph with vertices labeled as 1, 2, ..., k with the Seidel matrix SpGq rs ij s where s ij ¡1 if the vertices i and j are adjacent and s ij 1 if the vertices i and j are not adjacent and s ij 0 for the diagonal entries. Let H j be r j -regular and |V pH j q| n j , for every j 1, 2, ..., k. Let SL pH j q denote the Seidel signless Laplacian matrix of H j , 1, 2, ..., k.
where τ j °k i1 s ij n i . Theorem 6.2. Consider G rH 1 , H 2 , ..., H k s, where G is a simple connected graph with vertices labeled as 1, 2, ..., k and S rs ij s k¢k is the Seidel matrix of G and H j is r j -regular and |V pH j q| n j , for every j 1, 2, ..., k. Let tσ SL j1 2pn j ¡ 2r j ¡ 1q, σ SL j2 , ..., σ SL jnj u be the Seidel signless Laplacian eigenvalues of H j , for j 1, 2, ..., k. Then, the Seidel signless Laplacian spectrum of the G-join of the graphs H 1 , H 2 , ..., H k is given by,