Eigenvalues of zero-divisor graphs of finite commutative rings

We investigate eigenvalues of the zero-divisor graph $\Gamma(R)$ of finite commutative rings $R$ and study the interplay between these eigenvalues, the ring-theoretic properties of $R$ and the graph-theoretic properties of $\Gamma(R)$. The graph $\Gamma(R)$ is defined as the graph with vertex set consisting of all non-zero zero-divisors of $R$ and adjacent vertices $x,y$ whenever $xy = 0$. We provide formulas for the nullity of $\Gamma(R)$, i.e. the multiplicity of the eigenvalue 0 of $\Gamma(R)$. Moreover, we precisely determine the spectra of $\Gamma(\mathbb Z_p \times \mathbb Z_p \times \mathbb Z_p)$ and $\Gamma(\mathbb Z_p \times \mathbb Z_p \times \mathbb Z_p \times \mathbb Z_p)$ for a prime number $p$. We introduce a graph product $\times_{\Gamma}$ with the property that $\Gamma(R) \cong \Gamma(R_1) \times_{\Gamma} \ldots \times_{\Gamma} \Gamma(R_r)$ whenever $R \cong R_1 \times \ldots \times R_r.$ With this product, we find relations between the number of vertices of the zero-divisor graph $\Gamma(R)$, the compressed zero-divisor graph, the structure of the ring $R$ and the eigenvalues of $\Gamma(R)$.


Introduction
Let R be a finite commutative ring with 1 = 0 and let Z(R) denote its set of zero-divisors. As introduced by Anderson and Livingston [3] in 1999, the zerodivisor graph Γ(R) is defined as the graph with vertex set Z * (R) = Z(R)\{0} where two vertices x, y are adjacent if and only if xy = 0. The aim of considering these graphs is to study the interplay between graph theoretic properties of Γ(R) and the ring properties of R. In order to simplify the representation of Γ(R) it is often useful to consider the so-called compressed zero-divisor graph Γ E (R). This graph was first introduced by Mulay [9] and further studied in [13,16,2,11]. For an element r ∈ R let [r] R = {s ∈ R | ann R (r) = ann R (s)} and R E = {[r] R | r ∈ R}. Then Γ E (R) is defined as the graph Γ(R E ). Note that [0] R = {0}, [1] R = R\Z(R) and [r] R ⊆ Z(R)\{0} for every r ∈ R\([0] r ∪ [1] R ). The notations are adopted from Spiroff and Wickham [13].
The spectrum of a graph G is defined as the multi-set of eigenvalues, i.e. the roots of the characteristic polynomial of the adjacency matrix A(G). The aim of studying eigenvalues of graphs is to find relations between those values and structural properties of the graph. The author refers to [5] for a good introduction to spectral graph theory. The nullity η(G) of a graph G is defined as the multiplicity of the eigenvalue 0 of G. Obviously, we have that η(G) = dim A(G) − rank A(G). Background and further results on the nullity of graphs are summarized in [7]. Within spectral graph theory, most graphs are considered to be simple, i.e. to be undirected finite graphs without loops or multiple edges. By definition, Γ(R) has no multiple edges, and we can easily see that Γ(R) is undirected if and only if R is commutative. Moreover, as already proven by Anderson and Livingston [3,Theorem 2.2], the graph Γ(R) is finite if and only if R is finite or an integral domain. In the latter case, though, R has no zero-divisors at all and is just the empty graph. Hence, all our rings are assumed to be finite and commutative. However, in contrast to the original definition of Anderson and Livingston [3], we do not want to eliminate potential loops of our zero-divisor graphs since these loops provide important information about the structure of the ring R.
In order to determine the eigenvalues of a graph, it often can be useful to consider graph products. For example, in [1] the spectra of unitary Cayley graphs of finite rings could easily be determined by observing that these graphs are isomorphic to direct products of unitary Cayley graphs of finite local rings. The direct product G 1 ×G 2 of graphs G 1 and G 2 is defined as the graph with vertex set It is well-known that the adjacency matrix of the direct product G 1 × G 2 equals the Kronecker product A(G 1 ) ⊗ A(G 2 ). Therefore, if λ i resp. µ i are the eigenvalues of G 1 resp. G 2 , the eigenvalues of G 1 × G 2 are exactly the products λ i µ j . Moreover, the complete product • G 2 in order to make clear that the graphs were coalesced at v.
In this paper, we study the interplay between graph-theoretic properties of the zero-divisor graph Γ(R), the spectrum of Γ(R) and the ring properties of R. By now, surprisingly little is known about the eigenvalues and adjacency matrices of zero-divisor graphs. First research in this direction was done by Sharma et. al. [12] in 2011. They made some observations on the adjacency matrices and eigenvalues of the graphs Γ(Z p × Z p ) and Γ(Z p [i] × Z p [i]). Further results were found by Young [17] in 2015 and independently by Surendranath Reddy et. al. [14] in 2017. Both studied the graphs Γ(Z n ) and precisely determined the eigenvalues of Γ(Z p ), Γ(Z p 2 ), Γ(Z p 3 ) and Γ(Z p 2 q ) for p and q being prime numbers. Other recent papers on that topic are [15,10]. Note that in most of these papers the corresponding zero-divisor graphs were also considered with loops.
Our main approach is the following: since R is a finite ring, it can be written as R ∼ = R 1 × . . . × R r , where each R i is a finite local ring. A proof for this and further results within the theory of finite commutative rings can be found in [4]. In Section 2 we introduce a graph product x Γ with the property that With this graph product, in Section 3 we find a relation between the number of vertices of Γ E (R) and the property of R to be a local ring. Moreover, we derive formulas for the number of vertices of the zerodivisor graph Γ(R) resp. the compressed zero-divisor graph Γ E (R) in terms of the local rings R i . From these formulas, we can deduce a lower bound for the nullity of Γ(R). In Section 4 we restrict our considerations to rings which are isomorphic to direct products of rings of integers modulo n, i.e. R ∼ = Z p 1 t 1 × . . . × Z pr tr for (not necessarily distinct) prime numbers p i and positive integers r, t i . For these rings, we find the exact nullity of Γ(R) and present an easy approach to determine also the non-zero eigenvalues of Γ(R). For example, we precisely determine the spectra of Γ(Z p × Z p × Z p ) and Γ(Z p × Z p × Z p × Z p ) in terms of a prime number p. We also provide the characteristic polynomials of Γ(Z p 2 × Z p ) and Γ(Z p × Z p × Z q ) for primes q = p. This generalizes the results of Sharma et. al. [12], Young [17] and Surendranath Reddy et. al. [14].
Throughout this paper, we denote edges as sets of two vertices. For a graph G, we write A(G) for the adjacency matrix of G , V (G) for the set of vertices of G and χ G (x) = det(xI − A(G)) for the characteristic polynomial of G. If λ is an eigenvalue of G of multiplicity x, then we denote this by λ [x] . The number of elements in a set S is denoted by #S, and ϕ denotes Euler's totient function. For the set of units of a ring R, we write U (R).

Products of zero-divisor graphs
Let R ∼ = R 1 × . . . × R r be a ring, where each R i is a finite local ring. Note that in this case #R i = p t i i for some prime numbers p i and t i ∈ N. Our aim is to define a graph product × Γ such that our idea is the following: we first add the vertex 0 ∈ R i and the units of R i to the vertices of each zero-divisor graph Γ(R i ), as well as edges from 0 to every other vertex. Then, we take the direct product of these somehow extended zero-divisor graphs, each of which we will denote by EΓ(R i ), which yields the extended zero-divisor graph EΓ(R). Finally, by removing the vertex 0 ∈ R with all its edges, as well as all units of R, we end up with the zero-divisor graph Γ(R).
To formalize this, we define the unit graph U(R i ) of R i as the graph with vertex set U (R i ) and empty edge set. Moreover, let Z(R i ) resp. Z L (R i ) be the zero graph with vertex set {0} (where 0 ∈ R i ) and empty edge set resp. edge set {{0, 0}} (i.e. both graphs consist of one vertex only, and, in contrast to Z(R), the graph Z L (R) also has a loop at that vertex; we need this distinction for our result in Section 5). Now, the extended zero-divisor graph EΓ(R i ) is given by and we have that Hence, we define the associative product × Γ by where G\{v} denotes the graph G without the vertex v ∈ V (G) and all its adjacent edges. Note that Z( The product × Γ is illustrated in the following example: Figure 1 shows the zero-divisor graphs Γ(Z 8 ) and Γ(Z 4 ) and Figure 2 the extended zero-divisor graphs EΓ(Z 8 ) and EΓ(Z 4 ). In Figure 3 we see the direct product EΓ( The same also holds for the compressed zero-divisor graph, i.e. we have that In Section 5 we deduce a relation between the characteristic polynomial of Γ(R) and the one of the extended zero-divisor graph EΓ(R).

Nullity of zero-divisor graphs of finite commutative rings
The following theorem follows directly from the construction of the product × Γ : Then the number of zero-divisors of R, i.e. the number of vertices of the zero-divisor graph Γ(R) equals Proof. Since the R i 's are finite rings, each element of R i is either a zero-divisor or a unit. Thus, the elements of R iE are exactly the vertices of Γ E (R i ) together with [0] R i and [1] R i (since the elements of [1] R i are exactly the units of R i ). The statement follows from the construction of × Γ .
From this theorem, we can immediately deduce the following nice (algebraic) corollary: Corollary 3.3. If #V (Γ E (R)) + 2 is a prime number, then R is a local ring. Conversely, if R is a local ring, then #V (Γ E (R)) + 2 is a prime power.
Note that the converse of the second statement is not true in general, i.e. #V (Γ E (R)) + 2 being a prime power does not imply that R is a local ring, see . The corresponding compressed zero-divisor graph Γ E (R) has 5 vertices, see Figure 5. Corollary 3.3 shows that, therefore, R has to be a local ring.

Spectra of zero-divisor graphs of direct products of rings of integers modulo n
As already observed by Young [17], the adjacency matrix of the compressed zero-divisor graph is a so-called equitable partition of the adjacency matrix of Γ(R). A formal definition for this is given in [5]. We define the weighted adjacency matrix A(Γ E (R)) of the compressed zero-divisor graph as the matrix with (i, j)-th entry From [5, Lemma 2.3.1] it follows that every eigenvalue of A(Γ E (R)) is also an eigenvalue of A(Γ(R)). In general, it is not clear whether these eigenvalues are exactly the non-zero eigenvalues of Γ(R), i.e. whether A(Γ E (R)) always has full rank. But assuming that R is a product of rings of integers modulo n, we can prove the following: . . × Z pr tr for prime numbers p j and r, t j ∈ N. Then, rank A(Γ(R)) = rank A(Γ E (R)) = #V (Γ E (R)).
Proof. We can easily see that rank A(Γ(R)) = rank A(Γ E (R)) since for every r ∈ R, each element of [r] R contributes exactly the same row to the adjacency matrix A(Γ(R)). Thus, it suffices to show that rank A(Γ E (R)) = #V (Γ E (R)).
Obviously, this matrix has full rank #V (Γ E (R i )). Now, the graph EΓ E (R i ) arises from Γ E (R i ) by adding the vertices [1] R i and [0] R i and the edges With an appropriate enumeration of the vertices of EΓ E (R i ), it follows that the matrix A(EΓ E (R i )) equals This matrix has full rank, too. Since EΓ E (R) ∼ = EΓ E (R 1 ) × . . . × EΓ E (R r ), the matrix A(EΓ E (R)) equals the Kronecker product A(EΓ E (R 1 ))⊗. . .⊗A(EΓ E (R r )) which has the form  for non-zero entries x j . By the fact that the rank of the Kronecker product of two matrices equals the product of the ranks of these two matrices, we finally conclude that A(EΓ E (R)), and therefore also A(Γ E (R)) has full rank, i.e. rank A(Γ E (R)) = #V (Γ E (R)).
With this result, we are able to improve Theorem 3.5: Theorem 4.2. Let R ∼ = Z p 1 t 1 × . . . × Z pr tr for prime numbers p j and r, t j ∈ N. Then the zero-divisor graph Γ(R) has r i=1 (t i + 1) − 2 non-zero eigenvalues, and the nullity of Γ(R) equals Proof. By Theorem 4.1, the number of non-zero eigenvalues of Γ(R) equals the number of vertices of the compressed zero-divisor graph Γ E (R). Since Similar as in the proof of Theorem 3.5, we see that η(Γ(R)) = #V (Γ(R)) − #V (Γ E (R)). The number of units in . Thus, by Theorem 3.1, we get that and, therefore, the statement follows.
Note that the number of non-zero eigenvalues of Γ(Z p 1 t 1 × . . . × Z pr tr ) does not depend on the prime numbers p i but on the powers t i only. Now, we can easily determine the eigenvalues of Γ(R) for R ∼ = Z p 1 t 1 ×. . .×Z pr tr . Theorem 4.2 gives us the number of eigenvalues equal to zero. The nonzero eigenvalues can be computed as in the proof of Theorem 4.1. We illustrate this in the following examples. Note that the eigenvalues of the graphs Γ(Z p 2 ), Γ(Z p 3 ), Γ(Z p × Z q ) and Γ(Z p 2 × Z q ) for prime numbers p = q were already determined by Young [17] resp. Surendranath Reddy et. al. [14], and the ones of Γ(Z p × Z p ) by Sharma et. al. [12].
The ring Z p has no zero-divisors and, therefore, Γ(Z p ) is the empty graph. Thus, the matrix A(EΓ E (Z p )) is given by

Now, we compute the Kronecker product
which yields the matrix which has characteristic polynomial The roots of this polynomial, i.e. the non-zero eigenvalues of Γ(R), are and, therefore, the spectrum of Γ(R) equals spec(Γ(R)) = λ [2] 1 , λ [2] 2 , λ [1] 3 , λ Example 4.4. Let p be a prime number and R ∼ = Z p × Z p × Z p × Z p . By Theorem 4.2 the multiplicity of the eigenvalue 0 of Γ(R) equals Analogously as in Example 4.3, we find the matrix A(Γ E (R)) as a submatrix of the Kronecker product The characteristic polynomial of this matrix is and has roots λ 1 = (p − 1) 2 , λ 2 = −p 2 + p − 1, Hence, the spectrum of Γ(R) is given by spec(Γ(R)) = λ [5] 1 , λ [1] 2 , λ [3] 3 , λ [3] 4 , λ [1] 5 , λ Example 4.5. Unfortunately, if we consider not only products of the ring Z p but also of rings of the form Z p t for t > 1 or of the form Z q for a prime q = p, the eigenvalues of Γ(R) get very cumbersome. However, at least we want to include the characteristic polynomials of the graphs Γ(Z p × Z p × Z q ) and Γ(Z p 2 × Z p ).
Note that With the same method as in the examples before, we find the polynomials Remark 4.6. It is clear that two rings are isomorphic only if their respective zerodivisor graphs are isomorphic. Moreover, two graphs are isomorphic only if they have the same characteristic polynomial. Thus, in order to see that two rings are non-isomorphic, it might help to compare the characteristic polynomials of their corresponding zero-divisor graphs.

A relation between χ Γ(R) and χ EΓ(R)
The main interest in spectral graph theory of building graph products is that there are relations between the eigenvalues of graphs and the eigenvalues of their product. For example, we already mentioned that if λ i , µ i denote the eigenvalues of graphs G 1 resp. G 2 , then the eigenvalues of the direct product G 1 × G 2 are exactly the values λ i µ j . Similar results are also known for the point identification and the complete product of simple graphs: let G − v denote the graph arising from removing the vertex v of G together with all its edges, and let G be the complement of G (that is, the graph with same vertex set as G, where two distinct vertices are adjacent whenever they are non-adjacent in G), then the following holds: Lemma 5.1. Let G 1 and G 2 be simple graphs with v ∈ V (G 1 ) and w ∈ V (G 2 ). The point-identification v = w yields Lemma 5.2. Let G 1 and G 2 be simple graphs with #V (G 1 ) = n 1 and #V (G 2 ) = n 2 . Then the characteristic polynomial of the complete product of G 1 and G 2 equals Proofs for these lemmas are given in [6]. We can easily see that the formula in Lemma 5.1 still holds for graphs with loops if the graphs do not have loops on both vertices, v and w. Moreover, Hwang and Park [8] generalized the result of Lemma 5.2: Then Therefore, in the following let G denote the generalized complement of G, i.e. the graph with same vertex set as G, where two not necessarily distinct vertices are adjacent whenever they are non-adjacent in G. That is, the graph with adjacency matrix A(G) = J − A(G), where J denotes the all-1 matrix. Now, we are able to prove the following: Theorem 5.4. Let R be a finite commutative ring and n = #U (R), i.e. the number of units in R. Then we have that χ EΓ(R) (x) = x n−1 (−1) n+1 χ Γ(R) (−x)x + χ Γ(R) (x)(x 2 − n) .
We first determine the characteristic polynomial of U(R)∇ Z L (R) by applying Lemma 5.3 for A = 1 and B being the zero-matrix of dimension n × n. We can easily see that χ A (x) = x − 1, χÃ(x) = x, χ B (x) = x n and χB(x) = x n−1 (x − n). Thus, we get Analogously, we find the characteristic polynomial of Γ(R)∇ Z(R) for A = 0 and B = A(Γ(R)) to be χ Γ(R)∇ Z(R) (x) = (−1) n+1 χ Γ(R) (−x) + χ Γ(R) (x)(x + 1). Finally, with Lemma 5.1 we get Remark 5.5. If R ∼ = R 1 × . . . × R r , we can apply Theorem 5.4 to each of the rings R i , which gives us the characteristic polynomials χ EΓ(R i ) . By computing the roots of χ EΓ(R i ) , we find the eigenvalues of EΓ(R) to be all possible products of these roots, since EΓ(R) ∼ = EΓ(R 1 ) × . . . × EΓ(R r ). Unfortunately, it is difficult to extrapolate the eigenvalues of Γ(R) from the ones of EΓ(R), since the characteristic polynomial χ Γ(R) not only depends on χ EΓ(R) , but also on the characteristic polynomial of the generalized complement of Γ(R).