Laplacian spectrum of the unit graph associated to the ring of integers modulo

: Let R be a ring and U ( R ) be the set of unit elements of R . The unit graph G ( R ) of R is the graph whose vertices are all the elements of R , deﬁning distinct vertices x and y to be adjacent if and only if x + y ∈ U ( R ). The Laplacian spectrum of G ( Z n ) was studied when n = p m , where p is a prime and m is a positive integer. Consequently, in this paper, we study the Laplacian spectrum of G ( Z n ), for n = p 1 p 2 ... p k , where p i are distinct primes and i = 1 , 2 , ..., k .


Introduction
All graphs considered in this paper are finite, undirected, and may contain loops.Let G be a graph with vertex set V(G) = {v 1 , v 2 , ..., v n } and edge set E(G) = {e 1 , e 2 , ..., e m }.For 1 ≤ i, j ≤ n, two vertices v i and v j in G are adjacent (or neighbors) in G if v i and v j are endpoints of an edge e of G, and we write v i ∼ v j if v i is adjacent to v j in G.The degree of a vertex v in G, denoted by deg(v), is the number of edges incident with it, except that a loop at a vertex contributes twice to the degree of that vertex.The adjacency matrix of G is the n × n matrix A(G) = (a i j ), where a i j = 1 or 0 according to whether v i ∼ v j is in G or not.The Laplacian matrix L(G) of G is defined by L(G) := D(G) − A(G), where D(G) = Diag(d 1 , d 2 , ..., d n ) is the diagonal matrix such that d i are degrees of vertices of G. L(G) is a symmetric, real, and positive semidefinite matrix; all eigenvalues of L(G) are real and nonnegative.For a simple graph G, the sum of the entries in each row of L(G) is zero, and hence the smallest eigenvalue of L(G) is 0.More literature about the Laplacian matrix of graphs can be seen in [7,13].
The spectrum of a square matrix B, denoted by σ(B), is the multiset of all the eigenvalues of B.
For a graph G, the Laplacian spectrum of G is the spectrum of L(G), denoted by σ L (G).The Laplacian spectrum of graphs of rings has been widely studied in literature, see [3][4][5].
For a positive integer n, let Z n denote the ring of integers modulo n.In this paper, the elements of the ring Z n are referred to as 0, 1, 2, and n − 1.A nonzero element x ∈ Z n is a unit in Z n if x is relatively prime with n: (x, n) = 1.In 1990, the unit graph was first introduced by Grimaldi [8] for Z n as follows: the unit graph G(Z n ) is the graph obtained by setting all the elements of Z n to be vertices and defining distinct vertices x and y to be adjacent if and only if x + y is a unit in Z n .He discussed certain basic properties of the structure of the unit graph G(Z n ) and studied the covering number, the degree of a vertex, the independence number, the Hamilton cycles, and the chromatic polynomial of the graph G(Z n ).More about the unit graph G(Z n ) can be seen in [16,17].Later, Ashrafi et al. [2] generalized the unit graph from G(Z n ) to G(R) for an arbitrary ring R.They studied the chromatic index, diameter, girth, and planarity of G(R).In addition, they defined the closed unit graph Ḡ(R) by dropping the word "distinct" from the definition of the unit graph G(R), and stated that some of Ḡ(R)'s vertices may have loops.Many properties of the unit graph G(R) were investigated in [1,11,18].
The remaining parts of the paper are organized as follows: In Section 2, we present some preliminaries and deduce the Laplacian matrix of the direct product for graphs with loops.In Section 3, we obtain the Laplacian spectrum of the graphs G(Z pq ) and G(Z 2pq ), where p, q 2 are distinct primes.We deduce several consequences from these results, which include the determination of the Laplacian spectrum of G(Z n ) for n = p 1 p 2 ...p k , where p i are distinct primes and i = 1, 2, ..., k.

Laplacian matrix of the direct product of graphs with at most one loop at each vertex
Let us denote the graph with at most one loop at each vertex by Ḡ and let G be the simple graph corresponding to Ḡ.For v ∈ V(G), we denote by N(v) the set of all neighbors of v in G.Note that, if v has a loop, then v ∈ N(v) [15].For graphs G and Ḡ we define the following matrices: From Figure 1, we can find the following: , and In this section, we provide a formula for the Laplacian matrix of the direct product of graphs with at most one loop at each vertex.Let us first recall that the Kronecker product A ⊗ B of a p × q matrix A = (a i j ) by an r × s matrix B is defined as Other names for the Kronecker product are the tensor product or direct product.Let G 1 and G 2 be two graphs; the direct product of G 1 and G 2 is a graph, denoted by , and for which vertices (x 1 , y 1 ) and (x 2 , y 2 ) are adjacent if x 1 and x 2 are adjacent in G 1 , and y 1 and y 2 are adjacent in G 2 .Kaveh and Alinejad described the Laplacian matrix of G 1 ⊗ G 2 in terms of the Laplacian matrices of G 1 and G 2 , where G 1 and G 2 are simple graphs, as can be seen below [10].
The following lemma describes the degree matrix and the adjacency matrix of Ḡ1 ⊗ Ḡ2 .Lemma 2.1.Let Ḡ1 and Ḡ2 be graphs with at most one loop at each vertex.Then, Proof.(1) Suppose that Ḡ1 and Ḡ2 are two graphs with at most one loop at each vertex.If (x, y) ∈ V( Ḡ1 ⊗ Ḡ2 ), then N((x, y)) = N(x) × N(y) [12].It follows that N( Ḡ1 ⊗ Ḡ2 ) = N( Ḡ1 ) ⊗ N( Ḡ2 ).By using the fact that (x, y) has a loop in Ḡ1 ⊗ Ḡ2 if and only if x and y have a loop in Ḡ1 and Ḡ2 , respectively [9], then M( (2) For simple graphs G 1 and [6].If we agree on the convention that a 1 diagonal entry in the adjacency matrix of Ḡi , i = 1, 2, means a loop, whereas a 0 means no loop, then the adjacency matrix of Ḡ1 ⊗ Ḡ2 still corresponds to the Kronecker product of the adjacency matrices of A( Ḡ1 ) and A( Ḡ2 ).Thus, The following theorem generalizes the result about the Laplacian matrix of the direct product of simple graphs to the Laplacian matrix of the direct product of graphs with at most one loop at each vertex.
Theorem 2.1.Let Ḡ1 and Ḡ2 be graphs with at most one loop at each vertex.Then, the Laplacian matrix of (2.1) 1, and applying the properties of the direct product of matrices, we obtain Through basic cancellations, the result follows.
Rezagholibeigia et al. [14] observed that Ḡ(R × S ) Ḡ(R) ⊗ Ḡ(S ), where R × S is the direct product of the rings R and S .Motivated by their observations, we study the Laplacian matrix of G(Z pq ) G(Z p × Z q ) as the Laplacian matrix of Ḡ(Z p ) ⊗ Ḡ(Z q ) after removing the loops of this graph by deleting the matrix M( Ḡ(Z p )) ⊗ M( Ḡ(Z q )) from Eq (2.1).So, we have the following consequential result.
Corollary 2.1.The Laplacian matrix of G(Z pq ), where p, q 2 are primes, is

Laplacian spectrum of G(Z pq )
Recall that Euler's totient function, denoted by ϕ(n), is the number of positive integers less than or equal to n that are relatively prime to n.If p i , where i = 1, 2, ..., k, are primes, then ϕ(p i ) = p i − 1 and ϕ(p 1 p 2 ...p k ) = ϕ(p 1 )ϕ(p 2 )...ϕ(p k ).If 2 U(R), then x + x = 2x for all x ∈ R, and hence 2x U(R).So, in Ḡ(R), no vertex has a loop.That means that Ḡ(R) = G(R), which is pointed out in [2].If 2 ∈ U(R), then x + x = 2x is a unit in R for all x ∈ U(R) and hence x has a loop.So, in this case, the number of vertices in Ḡ(Z n ), which has a loop, is ϕ(n).The next result is derived from the previous discussion.
Proposition 3.1.The Laplacian matrix of G(Z 2p ), where p 2 is prime, is The following results will be used in the next part.
[2] Let R be a finite ring.Then, the following statements hold for the unit graph of R: ( Theorem 3.1.[16] Suppose q is an odd prime and suppose that n is a positive integer.Then, G(Z q n )'s Laplacian spectrum is given by Lemma 3.1.Let p, q, p i , where i = 1, 2, ..., k, be primes such that p, p i 2.Then, (1) N( Ḡ(Z q )) = ϕ(q)I, where I is a q × q identity matrix.
(2) N( Ḡ(Z 2p )) = ϕ(p)I, where I is a 2p × 2p identity matrix.Proof.(1) If q 2, then 2 ∈ U(Z q ), and hence all units of Z q have loops.The set of units in Z q is Z q − {0}.By Proposition 3.2, for v ∈ V( Ḡ(Z q )), such that 1 and hence N( Ḡ(Z q )) = ϕ(q)I.Now, if q = 2, then N(0) = {1} and N(1) = {0} in Ḡ(Z 2 ), and hence )), and hence N( Ḡ(Z 2p )) = ϕ(p)I. ( The proof is similar to that of 3.1. Theorem 3.2.If p, q 2 are primes, then the Laplacian spectrum of G(Z pq ) is Proof.By using Corollary 2.1 and Lemma 3.1, we have Now, to determine the spectrum of L(G(Z pq )), we suppose that X i and Y j are eigenvectors of L(G(Z p )) and L(G(Z q )) according to the eigenvalues λ i and µ j , respectively.That is, Therefore, the eigenvalues of L(G(Z pq )) are given by ϕ(p)µ j + λ i ϕ(q) − λ i µ j , where 1 ≤ i ≤ p and 1 ≤ j ≤ q.By using Theorem 3.1, we have So, the spectrum of L(G(Z pq )) consists of Hence, the Laplacian spectrum of G(Z pq ) is as in Eq (3.1).
Approaching the proof in a similar manner as with Theorem 3.2, the eigenvalues of L(G(Z 2pq )) are given by ϕ(p)µ j + λ i ϕ(q) − λ i µ j , where λ i , 1 ≤ i ≤ 2p, and µ j , 1 ≤ j ≤ q, are the eigenvalues of L(G(Z 2p )) and L(G(Z q )), respectively.Thus, the result follows from Theorems 3.1 and 3. .
The following theorem gives the Laplacian spectrum of G(Z n ) if n = p 1 p 2 ...p k , where p i are distinct primes and i = 1, 2, ..., k.Theorem 3.4.Let p i 2 be distinct primes and k be a positive integer, 1 ≤ i, j ≤ k.Then:

Figure 1 .
Figure 1.(a) The unit graph of the ring Z 3 .(b) The closed unit graph of the ring Z 3 .