Eigenvalues of zero-divisor graphs of finite commutative rings
- PDF / 711,873 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 106 Downloads / 206 Views
Eigenvalues of zero-divisor graphs of finite commutative rings Katja Mönius1 Received: 24 April 2020 / Accepted: 30 October 2020 © The Author(s) 2020
Abstract We investigate eigenvalues of the zero-divisor graph (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 (R). The graph (R) is defined as the graph with vertex set consisting of all nonzero zero-divisors of R and adjacent vertices x, y whenever x y = 0. We provide formulas for the nullity of (R), i.e., the multiplicity of the eigenvalue 0 of (R). Moreover, we precisely determine the spectra of (Z p ×Z p × Z p ) and (Z p ×Z p ×Z p ×Z p ) for a prime number p. We introduce a graph product × with the property that (R) ∼ = (R1 ) × · · · × (Rr ) whenever R ∼ = R1 × · · · × Rr . With this product, we find relations between the number of vertices of the zero-divisor graph (R), the compressed zero-divisor graph, the structure of the ring R and the eigenvalues of (R). Keywords EJMA-D-19-00287 · Zero-divisor graphs · Graph eigenvalues · Graph nullity · Graph products · Local rings
1 Introduction Let R be a finite commutative ring with 1 = 0 and let Z (R) denote its set of zerodivisors. As introduced by Anderson and Livingston [3] in 1999, the zero-divisor graph (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 x y = 0. The aim of considering these graphs is to study the interplay between graph theoretic properties of (R) and the ring properties of R. In order to simplify the representation of (R), it is often useful to consider the so-called compressed zero-divisor graph E (R). This graph was first introduced by Mulay [10] and further studied in [2,12,14,17].
B 1
Katja Mönius [email protected] Julius-Maximilians-Universitat Wurzburg, Würzburg, Germany
123
Journal of Algebraic Combinatorics
Definition 1.1 (Compressed zero-divisor graph) For an element r ∈ R let [r ] R = {s ∈ R | ann R (r ) = ann R (s)} and R E = {[r ] R | r ∈ R}. Then, the compressed zero-divisor graph E (R) is defined as the graph (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 [14]. The spectrum of a graph G is defined as the multiset 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 [5,7] for good introductions to graph theory and spectral graph theory, respectively. The nullity η(G) of a graph G is defined as the multiplicity of the eigenvalue 0 of G. It is easy to see that η(G) = dim A(G) − rank A(G), where dim A(G) denotes the dimension of the domain of the linear transformation associated to the matrix A(G), i.e., the number of columns of A(G). Background and further results on the
Data Loading...
