Laplacian spectrum of comaximal graph of the ring ℤ_n

1 Introduction

Let G be a finite simple undirected graph with vertex set V(G)={v1,v2,…,vn}. If two vertices v1,v2 are adjacent, we denote it by vi∼vj. The join of two graphs G1=(V1,E1) and G2=(V2,E2) denoted by G1∨G2 is a graph obtained from G1 and G2 by joining each vertex of G1 to all vertices of G2. The union of two graphs G1=(V1,E1) and G2=(V2,E2) denoted by G1∪G2 is a graph with vertex set V=V1∪V2 and edge set E=E1∪E2. The adjacency matrix of G denoted by A(G)=(aij) is an n×n matrix defined as aij=1 when vi∼vj and 0 otherwise. The Laplacian matrix L(G) of G is defined as L(G)=D(G)−A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of vertex degrees. Since the matrix L(G) is a real, symmetric, and a positive semi-definite matrix, all its eigenvalues are real and non-negative. Also 0 is an eigenvalue of L(G) with eigenvector [1,1,1,…,1]T whose multiplicity equals the number of connected components in the graph G. Let the eigenvalues of L(G) be denoted by λ1≥λ2≥⋯≥λn−1≥λn=0. The largest eigenvalue λ1 is known as the spectral radius of G, and the second smallest eigenvalue λn−1 is known as the algebraic connectivity of G. Also λn−1>0 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)=min{∣S∣:Sis a separating set ofG}. The papers [1] and [2] list several interesting properties of λn−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 Γ(R) was introduced by Sharma and Bhatwadekar in [5]. The vertices of Γ(R) are the elements of the ring R, and two distinct vertices x,y of Γ(R) 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 Γ(R) denoted by χ(Γ(R)) is finite. It was further shown that χ(Γ(R)) satisfies χ(Γ(R))=t+l, 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 Γ(R), readers may refer to the works [6,7] and [8]. In this paper, the Laplacian spectrum of the comaximal graph of the finite ring Zn, denoted by Γ(Zn), 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(Zn) and investigate some structural properties of Γ(Zn) and find the characteristic polynomial of Γ(Zn) for n>2. We then explicitly determine the spectrum of Γ(Zn) for n=pαqβ, where p,q are distinct primes and α,β are non-negative integers and conclude that Γ(Zn) is Laplacian integral for n=pαqβ. In Section 4, we discuss the vertex and the algebraic connectivity of Γ(Zn). In Section 5, we find an upper bound on the second largest eigenvalue of Γ(Zn) 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(Zn). We also determine the multiplicity of the algebraic connectivity of Γ(Zn). In Section 6, we study an induced subgraph of Γ(Zn) formed by the non-zero non-unit elements of Zn. Finally, in Section 7, we provide some problems for further research.

2 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, λi(G) shall denote the ith eigenvalue of L(G).

3 Structure of Γ(Zn)

We denote the elements of the ring Zn by {0,1,2,…,n−2,n−1}. If x∈Zn, 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(Zn). We show that Γ(Zn) can be expressed as the join and union of certain induced subgraphs of Γ(Zn). We then investigate the Laplacian spectra of Γ(Zn) for various n. We first find an equivalent condition for adjacency of two vertices in Γ(Zn).

By using the adjacency criterion for any two vertices in Γ(R), we find that two vertices vi,vj∈Γ(Zn) are adjacent if and only if Znvi+Znvj=Zn. Now since Zn is a principal ideal ring (PIR), so Znvi=⟨vi⟩=⟨gcd(vi,n)⟩. Since sum of two ideals is again an ideal, so the adjacency criterion in Γ(Zn) becomes the following:

We have V(Γ(Zn))=S∪T, where S={a:gcd(a,n)=1} and T=Zn⧹S. Clearly, for all v∈S, deg(v)=n−1, and deg(0)=φ(n). Let G1 denote the induced subgraph of Γ(Zn) on the set S and G2′ denote the induced subgraph of Γ(Zn) on the set T. We have,

Again, if we let G2 to be the induced subgraph of Γ(Zn) on the set T⧹{0}, then

We can make the following observations as applications of equation (3).

The following observations about μ(Γ(Zn)) are evident:

From equation (4) of Theorem 3.3, we find that the eigenvalues of Γ(Zn) are known if the spectrum of G2 given in equation (3) is completely determined. We thus proceed to study the graph G2 in more detail.

3.1 Structure of G2

Let n=p1α1p2α2⋯pkαk be a prime factorization of n, where p1<p2<⋯<pk are primes and αi are positive integers. We say that d is a proper divisor of n if d divides n and d∉{1,n}. The total number of positive divisors of n equals (α1+1)(α2+1)⋯(αk+1). The total number of proper positive divisors of n will be given by w=(α1+1)(α2+1)⋯(αk+1)−2.

Let d1<d2<⋯<dw be the set of all proper divisors of n arranged in increasing order. For each di, where 1≤i≤w, we define

(5)Adi={x:gcd(x,n)=di}.

Any element of Adi is of the form zdi, where gcdz,ndi=1 and hence the number of elements of Adi is φndi. Thus, ∣Adi∣=φndi. Clearly, V(G2)=⋃i=1wAdi.

Lemma 3.6

xi∈Adiis adjacent toxj∈Adjif and only ifgcd(di,dj)=1.

Proof

Assume that xi∈Adi is adjacent to xj∈Adj. Using equation (2), xi adjacent to xj implies that either gcd(xi,xj)=1 or gcd(xi,xj) is a unit in Zn. We consider the following two cases:

1. gcd(xi,xj)=1. Let gcd(di,dj)=d, then d∣di,d∣dj. Since gcd(xi,n)=di and gcd(xj,n)=dj, we have di∣xi and dj∣xj, which in turn implies that d∣xi and d∣xj. Again since gcd(xi,xj)=1, d=1 and hence gcd(di,dj)=1.

2. gcd(xi,xj) is a unit in Zn. Let gcd(xi,xj)=a which is a unit in Zn and hence, gcd(a,n)=1. Let gcd(di,dj)=d, then d∣di,d∣dj. Since gcd(xi,n)=di and gcd(xj,n)=dj, we have di∣xi,di∣n and dj∣xj,dj∣n, which in turn implies that d∣xi, d∣xj and d∣n. Since gcd(xi,xj)=a, d∣a. Since gcd(a,n)=1, from the facts that d∣a and d∣n, it follows that d=1. Hence, gcd(di,dj)=1.

Thus if xi∈Adi is adjacent to xj∈Adj, then gcd(di,dj)=1.

Conversely, we now assume that gcd(di,dj)=1. Let d=gcd(xi,xj). We claim that either d=1 or d is a unit in Zn. Assume the contrary, then d>1 and d is not a unit in Zn, which implies d(>1) divides n.

Ifd=gcd(xi,xj),thend∣xi,d∣xj⇒(d∣xi,d∣n)and(d∣xj,d∣n)⇒d∣di=gcd(xi,n)andd∣dj=gcd(xj,n)⇒d∣gcd(di,dj)=1,which is a contradiction.

Thus, either d=1 or d is a unit in Zn, and hence, ⟨xi⟩+⟨xj⟩=⟨d⟩=Zn which implies by equation (2) that xi∈Adi is adjacent to xj in Adj.□

Lemma 3.7

Ifvi∈Adiis adjacent tovj∈Adjfor somei≠j, thenviis adjacent tovjfor allvj∈Adj.

Proof

Let vi∈Adi is adjacent to vj∈Adj for some i≠j, then using Lemma 3.6gcd(di,dj)=1. Let vj′≠vj be another member of Vj, then gcd(vj′,n)=dj. Using the fact that gcd(di,dj)=1 and Lemma 3.6, we conclude that vj′ is adjacent to vi.□

Lemma 3.8

No two members of the setAdiare adjacent.

Proof

If vi,vj∈Adi, then gcd(vi,n)=gcd(vj,n)=di. Using Lemma 3.6, the proof follows.□

If vi∈Adi, using Lemma 3.7, we observe that the number of neighbors of vi in Adj, where j≠i is fixed, i.e., either the number of neighbors of vi in Adj equals 0 or ∣Adj∣. Also using Lemma 3.8, the number of neighbors of vi in Adi equals 0 for all 1≤i≤w. If we denote Vi=Adi, where 1≤i≤w, then using Definition 2.5, we find that V1∪V2∪⋯∪Vw is an equitable partition of graph G2.

Thus, we have the following theorem:

Theorem 3.9

For anyn≥2, the induced subgraphG2ofΓ(Zn)with vertex setV(G2)has an equitable partition asV(G2)=⋃i=1wAdi, wherewdenotes the total number of positive proper divisors ofnand the setsAdihave been defined as in equation (5).

Using Theorems 3.9 and 2.7, it is evident that G2 is the H-join of the graphs Gdi, where Gdi is the induced subgraph of Γ(Zn) on Adi, and H can be obtained as follows:

Construction of H:

V(H)={di:1≤i≤w,wherediis a positive proper divisor ofn}. The vertices di,dj are adjacent in H if and only if gcd(di,dj)=1. Thus, E(H)={didj:gcd(di,dj)=1}.

We use Theorem 2.8 to determine the spectrum of G2. We find that G2 is the H- join of Gdi, where Gdi is a null graph on φndi vertices. Hence, σ(Gdi)={0}. Also,

NH(dj)={di:gcd(di,dj)=1},

and hence,

Ndj=∑di∈NH(dj)ni=∑di:gcd(di,dj)=1φndi.

Moreover, ndi=φndi, where 1≤i≤w.

Figure 1

H for n=pqr.

Example 3.10

If n=pqr, where p,q,andr, are primes with p<q<r, then the proper positive divisors of n are p,q,r,pq,pr,qr. Using the construction of H given earlier, we find that G2 is the H join of Gp,Gq,Gr,Gpq,Gpr,Gqr, where H is given by Figure 1.

Now we have,

(6)Np=φpqrq+φpqrr+φpqrqr=φ(pr)+φ(pq)+φ(p)=(p−1)(r−1)+(p−1)(q−1)+(p−1)=(p−1){r−1+q−1+1}=(p−1)(q+r−1).Similarly,Nq=(q−1)(p+r−1),Nr=(r−1)(p+q−1)Npq=(p−1)(q−1),Npr=(p−1)(r−1)andNqr=(q−1)(r−1).

Also

(7)np=(q−1)(r−1),nq=(p−1)(r−1),nr=(p−1)(q−1)npq=r−1,npr=q−1,nqr=p−1.

Using Theorem 2.8, we find that the eigenvalues of G2 are (p−1)(q+r−1) with multiplicity qr−r−q, (q−1)(p+r−1) with multiplicity pr−r−p, (r−1)(p+q−1) with multiplicity pq−p−q, (p−1)(q−1) with multiplicity r−2, (p−1)(r−1) with multiplicity q−2, and (q−1)(r−1) 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(Zn) for n=pαqβ, where p,q are primes, and α,β are nonnegative integers.

Theorem 3.11

Whenn=pm, wherepis a prime andm>1is a positive integer, then the eigenvalues ofΓ(Zn)arenwith multiplicityφ(n), φ(n)with multiplicityn−φ(n)−1and 0 with multiplicity 1.

Proof

When p is a prime and m>1 is a positive integer, the proper divisors of pm are p,p2,p3,…,pm−2,pm−1. We partition the vertex set V(G2) of G2 as V1,V2,…,Vm−2,Vm−1, where Vi=Api={x:gcd(x,n)=pi}.

Since gcd(pi,pj)=pmin{i,j}≠1, using Lemmas 3.6 and 3.8, we find that xi∈Vi is not adjacent to xj∈Vj for all 1≤i,j≤m−1.

Thus, no two vertices in the graph G2 are adjacent and hence G2=K¯n−φ(n)−1. By using equation (4), we obtain μ(Γ(Zn))=x(x−n)φ(n)(x−φ(n))n−φ(n)−1.□

Theorem 3.12

Ifn=pαqβ, wherep,qare primes withp<qandα,βare positive integers, then the eigenvalues ofΓ(Zn)arenwith multiplicityφ(n), (t+1)(p−1)+φ(n)with multiplicity(t+1)(q−1)−1, (t+1)(q−1)+φ(n)with multiplicity(t+1)(p−1)−1, φ(n)with multiplicityt+1, and(t+1)(p+q−2)+φ(n),0each with multiplicity 1, wheret=pα−1qβ−1−1.

Figure 2

G2 for n=pαqβ.

Proof

If n=pαqβ where p,q are primes with p<q and α,β are positive integers, then the proper divisors of n are piqj, where 0≤i≤α, 0≤j≤β with i+j∉{0,α+β}. We partition the vertex set V(G2) as follows:

(8)V(G2)=(Ap∪Ap2∪⋯∪Apα)∪(Aq∪Aq2∪⋯∪Aqβ)∪(∪j=1βApqj)∪(∪j=1βAp2qj)∪⋯∪(∪j=1β−1Apαqj).

If 1≤i≤α, 1≤j≤β, then gcd(pi,qj)=1. Using Lemmas 3.6 and 3.7, we find that every vertex of Api is adjacent to every vertex of Aqj.

Also Lemma 3.6 indicates that if 1≤i≤α, 1≤j≤β with i+j≠α+β, then no vertex of Apiqj is adjacent to any other vertex of G2. If we draw the graph G2 with the vertex partitions as given in equation (8), it looks like Figure 2. (A solid line in the figure indicates that each vertex of Adi is adjacent to each vertex of Adj. No line between two nodes Adi and Adj indicates that no vertex of Adi is adjacent to any vertex of Adj.)

Let G21 be the induced subgraph of G2 on the set Ap∪Ap2∪⋯∪Apα and G22 be the induced subgraph of G2 on the set Aq∪Aq2∪⋯∪Aqβ.

Now the number of elements in Ap∪Ap2∪⋯∪Apα is ∑i=1α∣Api∣. Hence,

(9)∑i=1α∣Api∣=∣Ap∣+∣Ap2∣+⋯+∣Apα∣=φ(pα−1qβ)+φ(pα−2qβ)+⋯+φ(qβ)=pα−11−1pqβ1−1q+pα−21−1pqβ1−1q+⋯+p1−1pqβ1−1q+qβ1−1q=qβ1−1q1−1p{p+p2+⋯+pα−1}+qβ1−1q=qβ1−1q1−1pp(pα−1−1)p−1+qβ1−1q=qβ1−1q(pα−1−1+1)=qβ−1pα−1(q−1).

Again the number of elements in the set Aq∪Aq2∪⋯∪Aqβ is ∑i=1β∣Aqi∣. By using similar calculations as in equation (9), we find that

(10)∑i=1β∣Aqi∣=pα−1qβ−1(p−1).

The vertices of G2 that are not adjacent to any other vertex in G2 are the members of the set (⋃j=1βApqj)∪(⋃j=1βAp2qj)∪⋯∪(⋃j=1β−1Apαqj). By using equations (9) and (10), the number of such vertices denoted by t equals

t=pαqβ−φ(pαqβ)−1−pα−1qβ−1(p−1)−pα−1qβ−1(q−1)=pαqβ−1−pα−1qβ−1{(p−1)(q−1)+p−1+q−1}=pαqβ−1−pα−1qβ−1(pq−1)=pα−1qβ−1−1.

Clearly, the induced subgraph of G2 on pα−1qβ−1−1 vertices is a null graph. Since every vertex of the graph G21 is adjacent to every vertex of the graph G22 and the remaining vertices of G2 are not adjacent to any other vertex, the following is evident

(11)G2=(G21∨G22)∪K¯t.

By using equations (9) and (10) and Theorem 2.2, we obtain

(12)μ((G21∨G22),x)=x(x−pα−1qβ−1(q−1))pα−1qβ−1(p−1)−1(x−pα−1qβ−1(p−1))pα−1qβ−1(q−1)−1(x−(pα−1qβ−1(p+q−2))).

By using Theorem 2.3, equations (11) and (12) we obtain,

(13)μ(G2,x)=xt×μ((G21∨G22),x)=xt+1(x−pα−1qβ−1(q−1))pα−1qβ−1(p−1)−1(x−pα−1qβ−1(p−1))pα−1qβ−1(q−1)−1(x−(pα−1qβ−1(p+q−2))).

By using equation (13) in equation (4), we have

μ(Γ(Zn),x)=x(x−n)φ(n)μ(G2,x−φ(n))=x(x−n)φ(n)(x−φ(n))t+1(x−pα−1qβ−1(q−1)−φ(n))pα−1qβ−1(p−1)−1×(x−pα−1qβ−1(p−1)−φ(n))pα−1qβ−1(q−1)−1(x−(pα−1qβ−1(p+q−2)−φ(n))).

Thus, the eigenvalues of Γ(Zn) are n with multiplicity φ(n), (t+1)(p−1)+φ(n) with multiplicity (t+1)(q−1)−1, (t+1)(q−1)+φ(n) with multiplicity (t+1)(p−1)−1, φ(n) with multiplicity t+1, and (t+1)(p+q−2)+φ(n),0 each with multiplicity 1.□

By using Corollary 3.5 and Theorems 3.11 and 3.12, the following is evident.

Theorem 3.13

Ifn=pαqβ, wherep,qare primes andα,βare non-negative integers, thenΓ(Zn)is Laplacian integral.

By using Theorem 3.12 and Kirchhoff’s matrix tree theorem [10], we have

Theorem 3.14

Ifn=pαqβ, wherep,qare primes andα,βare non-negative integers, then the number of spanning trees ofΓ(Zn)is expressed as follows:

st(Γ(Zn))=((t+1)pq)t(p−1)(q−1)((t+1)q(p−1))(t+1)(q−1)−1((t+1)p(q−1))(t+1)(p−1)−1×((t+1)(p−1)(q−1))t+1((t+1)(pq−1))wheret=pα−1qβ−1−1.