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Abstract: In this paper, we study the interplay between the structural and spectral properties of the comaximal graph Γ n ( ) of the ring n for n 2 > . We first determine the structure of Γ n ( ) and deduce some of its properties. We then use the structure of Γ n ( ) to deduce the Laplacian eigenvalues of Γ n ( ) for various n. We show that Γ n ( ) is Laplacian integral for n p q α β = , where p q , are primes and α β , are nonnegative integers and hence calculate the number of spanning trees of Γ n ( ) for n p q α β = . The algebraic and vertex connectivity of Γ n ( ) have been shown to be equal for all n. An upper bound on the second largest Laplacian eigenvalue of Γ n ( ) has been obtained, and a necessary and sufficient condition for its equality has also been determined. Finally, we discuss the multiplicity of the Laplacian spectral radius and the multiplicity of the algebraic connectivity of Γ n ( ). We then investigate some properties and vertex connectivity of an induced subgraph of Γ n ( ). Some problems have been discussed at the end of this paper for further research.
Keywords: comaximal graph, Laplacian eigenvalues, vertex connectivity, algebraic connectivity, Laplacian spectral radius, finite ring . The largest eigenvalue λ 1 is known as the spectral radius of G, and the second smallest eigenvalue λ n 1 − is known as the algebraic connectivity of G. Also λ 0 n 1 > − if and only if G is connected. The term algebraic connectivity was given by Fiedler in [1]. A separating set in a connected graph G is a set S V G ( ) ⊂ such that V G S ( )⧹ has more than one connected component. The vertex connectivity of G denoted by κ G ( ) is defined as κ G S S G min : is a separating set of ( ) {| | } = . The papers [1] and [2] list several interesting properties of λ n 1 − and κ. Readers may refer to [3] for a survey on the Laplacian matrix of a graph G. A graph G is called Laplacian integral if all the eigenvalues of L G ( ) are integers. We follow [4] for definitions of standard terms in graph theory.
Let R be a commutative ring with unity 1 0 ≠ . The comaximal graph of a ring R denoted by R Γ( ) was introduced by Sharma and Bhatwadekar in [5]. The vertices of R Γ( ) are the elements of the ring R, and two distinct vertices x y , of R Γ( ) are adjacent if and only if Rx Ry R + = . They proved that R is a finite ring if and only if the chromatic number of R Γ( ) denoted by χ R Γ ( ( )) is finite. It was further shown that χ R Γ ( ( )) satisfies χ R t l Γ ( ( )) = + , where t denotes the number of maximal ideals of R and l denotes the number of units of R. A lot of research has been done on the comaximal graph of a ring R over the last few decades. For some literature on R Γ( ), readers may refer to the works [6,7] and [8]. In this paper, the Laplacian spectrum of the comaximal graph of the finite ring n , denoted by Γ n ( ), has been studied for various n. In Section 2, we provide the preliminary theorems that have been used throughout the paper. In Section 3, we discuss the structure of Γ n ( ) and investigate some structural properties of Γ n ( ) and find the characteristic polynomial of Γ n ( ) for n 2 > . We then explicitly determine the spectrum of Γ n ( ) for n p q α β = , where p q , are distinct primes and α β , are non-negative integers and conclude that Γ n ( ) is Laplacian integral for n p q α β = . In Section 4, we discuss the vertex and the algebraic connectivity of Γ n ( ). In Section 5, we find an upper bound on the second largest eigenvalue of Γ n ( ) and determine a necessary and sufficient condition when it attains its bounds. We use it to determine the multiplicity of the spectral radius of Γ n ( ). We also determine the multiplicity of the algebraic connectivity of Γ n ( ). In Section 6, we study an induced subgraph of Γ n ( ) formed by the non-zero non-unit elements of n . Finally, in Section 7, we provide some problems for further research.

Preliminaries
In this section, we will provide some preliminary theorems that will be required in our subsequent sections. Throughout this paper by eigenvalues and characteristic polynomial of a given graph G, we shall mean the eigenvalues and characteristic polynomial of the Laplacian matrix L G ( ) of G. Also, the characteristic polynomial and the multiset of eigenvalues of G have been denoted by μ G x , ( ) and σ G ( ), respectively. Thus, λ G i ( ) shall denote the ith eigenvalue of L G ( ). ∨ denote the join of two graphs G 1 and G 2 . Then, where n 1 and n 2 are orders of G 1 and G 2 respectively. . Then, , .
is given by 3 Structure of Γ n ( ) We denote the elements of the ring n by n n 0, 1, 2, , 2, , then x ⟨ ⟩ will denote the ideal generated by x. We follow [14] for standard definitions in ring theory.
In this section, we describe the structure of Γ n ( ). We show that Γ n ( ) can be expressed as the join and union of certain induced subgraphs of Γ n ( ). We then investigate the Laplacian spectra of Γ n ( ) for various n. We first find an equivalent condition for adjacency of two vertices in Γ n ( ). By using the adjacency criterion for any two vertices in R ⟩ . Since sum of two ideals is again an ideal, so the adjacency criterion in Γ n ( ) becomes the following: Let G 1 denote the induced subgraph of Γ n ( ) on the set and G 2 ′ denote the induced subgraph of Γ n ( ) on the set . We have, Again, if we let G 2 to be the induced subgraph of Γ n ( ) on the set 0 { } ⧹ , then We can make the following observations as applications of equation (3).
Proof. By using equation (3) and Theorems 2.2 and 2.3, we obtain, The following observations about μ Γ n ( ( )) are evident: Corollary 3.4. If n 2 > , then n is an eigenvalue of Γ n ( ) with multiplicity at least φ n ( ).
Corollary 3.5. If n p = , where p is a prime number, then p and 0 are eigenvalues of Γ n ( ) with multiplicity p 1 − and 1, respectively.
From equation (4) of Theorem 3.3, we find that the eigenvalues of Γ n ( ) are known if the spectrum of G 2 given in equation (3) is completely determined. We thus proceed to study the graph G 2 in more detail.

Structure of G 2
Let n p p p α α k α 1 2 . The total number of positive divisors of n equals α α α The total number of proper positive divisors of n will be given by w α α α ) is a unit in n . We consider the following two cases: which is a unit in n and hence, a n gcd , Since a n gcd , 1 ( ) = , from the facts that d a | and d n | , it follows that d 1 Thus Conversely, we now assume that . We claim that either d 1 = or d is a unit in n . Assume the contrary, then d 1 > and d is not a unit in n , which implies d 1 ( ) > divides n.   Using Theorems 3.9 and 2.7, it is evident that G 2 is the H -join of the graphs G di , where G di is the induced subgraph of Γ n ( ) on A di , and H can be obtained as follows: Also n q r n p r n p q n r n q n p Using Theorem 2.8, we find that the eigenvalues of G 2 are p q r − with multiplicity p 2 − and remaining eigenvalues are the eigenvalues of 6 6 × matrix M (equation (1)) whose entries can be determined from equations (6) and (7).
We now find the spectrum of Γ n ( ) for n p q α β = , where p q , are primes, and α β , are nonnegative integers.
Theorem 3.11. When n p m = , where p is a prime and m 1 > is a positive integer, then the eigenvalues of Γ n ( ) are n with multiplicity φ n ( ), φ n ( ) with multiplicity n φ n 1 ( ) − − and 0 with multiplicity 1.
Proof. When p is a prime and m 1 > is a positive integer, the proper divisors of p m are p p p , , , , Since We partition the vertex set V G 2 ( ) as follows: Using Lemmas 3.6 and 3.7, we find that every vertex of A p i is adjacent to every vertex of A q j .
Also Lemma 3.6 indicates that if i α 1 ≤ ≤ , j β 1 ≤ ≤ with i j α β + ≠ + , then no vertex of A p q i j is adjacent to any other vertex of G 2 . If we draw the graph G 2 with the vertex partitions as given in equation (8)

Now the number of elements in
Again the number of elements in the set By using similar calculations as in equation (9), we find that The vertices of G 2 that are not adjacent to any other vertex in G 2 are the members of the set . By using equations (9) and (10) Clearly, the induced subgraph of G 2 on p q 1 vertices is a null graph. Since every vertex of the graph G 21 is adjacent to every vertex of the graph G 22 and the remaining vertices of G 2 are not adjacent to any other vertex, the following is evident By using equations (9) and (10) By using equation (13) in equation (4) Thus, the eigenvalues of Γ n ( ) are n with multiplicity φ n ( ), t p φ n

Algebraic connectivity and vertex connectivity of Γ n ( )
In this section, we investigate the algebraic connectivity (λ n 1 − ) and vertex connectivity (κ) of Γ n ( ) for any n 2 > . We also show that λ n 1 − and κ are equal for any n 2 > .
Proof. Since 0 is always an eigenvalue of the Laplacian matrix of a given graph G, so the Laplacian matrix of the graph G 2 also has 0 as an eigenvalue. By using equation where g x ( ) is a polynomial of degree n φ n 2 ( ) − − . Hence, φ n ( ) is an eigenvalue of Γ n ( ) with multiplicity at least 1. Proof. Using Lemma 4.1, φ n ( ) is an eigenvalue of Γ n ( ). Since the smallest root of the polynomial g x φ n ( ( )) − in equation (14) is φ n ( ) and φ n n 0 ( ) < < , we conclude that the second smallest root of μ x Γ , n ( ( ) ) is φ n ( ), which implies that λ φn Γ Proof. By using equation (3), we find that , we find that 1 is a disconnected graph on n φ n ( ) − vertices and 2 is a graph on φ n ( ) vertices. Clearly, λ λ K φn . We find that if we assume κ φ n Γ n ( ( )) ( ) = , then all the conditions of Theorem 2.6 along with the inequality λ κG n 2 In this section, we discuss the second largest eigenvalue λ 2 of Γ n ( ), which in turn helps us to find certain information about the largest eigenvalue λ 1 of Γ n ( ). We first study the connectivity of G 2 . We first assume that G 2 is connected. To show that n is a product of distinct primes, we prove that α 1 i = for all i m 1 ≤ ≤ . Assume the contrary that α 1 i > for at least one i. Without loss of generality, we take α 1 1 > . We consider the vertex a p p p p m Clearly, a 0 ≠ as α 1 1 > . Consider any other vertex of G 2 say w.
where A di has been defined in equation (5) ≠ , by using Lemma 3.6, we conclude that w is not adjacent to a. Since w is arbitrary, we find that the vertex a G 2 ∈ is not adjacent to any other vertex of G 2 , which contradicts the fact that G 2 is connected. Hence, our assumption that α 1 1 > is false. Thus, α 1 i = for all i m 1 ≤ ≤ , which proves that n is a product of distinct primes. Conversely, we assume that n is a product of distinct primes. To show that G 2 is connected, we choose two arbitrary distinct vertices x x G ,

∈
, then x p1 is adjacent to x p2 . Thus, using Lemma 3.6, we obtain a path of length 3 from x i to x j given by x x x x~ĩ p p j 1 2 . By combining cases 1 and 2, we find that any two vertices of G 2 are either adjacent or there exists a path between them, which implies that G 2 is connected when n is a product of distinct primes. Thus, G 2 is connected if and only if n is a product of distinct primes. □ We now investigate the connectivity of G 2 when n is a product of distinct primes.
When n is a product of two distinct primes, i.e., n pq = , then n has only two distinct proper positive divisors namely p and q. Thus V G A A p q 2 ( ) = ∪ . Since p q gcd , 1 ( ) = , using Lemma 3.6, G 2 becomes as shown in Figure 3.
(Here, the solid line indicates that each vertex of A p is adjacent to each vertex of A q ).
Clearly, G 2 is disconnected when n pq = . In the next theorem, we investigate the connectivity of G 2 when n is a product of more than two distinct primes.
Theorem 5.2. If n is a product of more than two distinct primes, then G 2 is connected. Then, x A We now discuss the vertex connectivity of G 2 . Since G 2 is connected if and only if n is a product of distinct primes, we discuss κ G 2 ( ) when n p p p m

( ) ( ) ≤ =
We prove the aforementioned formally in the following theorem: