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On the adjacency spectrum of zero divisor graph of ring \(\mathbb{Z}_n\). (English) Zbl 1497.05143

Summary: The zero divisor graph \({\Gamma}(R)\) of a commutative ring \(R\) with unity is a simple undirected graph whose vertices are all nonzero zero divisors of \(R\) and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(xy=0\). In this paper, we study the graphical structure and the adjacency spectrum of the zero divisor graph of ring \(\mathbb{Z}_n\). For any non-prime positive integer \(n\geq 4\) with \(\xi\) number of proper divisors, we show that the adjacency spectrum of \({\Gamma}(\mathbb{Z}_n)\) consists of the eigenvalues of a symmetric matrix \(C({\Upsilon}_n)\) of size \(\xi\times\xi \), and at the most \(0\) and \(-1\). Also, we find the exact multiplicity of the eigenvalue \(0\) and show that all eigenvalues of \(C({\Upsilon}_n)\) are nonzero, by determining the rank and nullity of the adjacency matrix of \({\Gamma}(\mathbb{Z}_n)\). We find the values of \(n\) for which the adjacency spectrum of \({\Gamma}(\mathbb{Z}_n)\) contains only nonzero eigenvalues. Finally, by computing the characteristic polynomial of the matrix \(C({\Upsilon}_n)\), we determine the characteristic polynomial of \({\Gamma}(\mathbb{Z}_n)\) whenever \(n\) is a prime power.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C31 Graph polynomials
05C75 Structural characterization of families of graphs
13A99 General commutative ring theory
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