On the cozero-divisor graphs associated to rings

Abstract

Let R be a ring with unity. The cozero-divisor graph of a ring R, denoted by Γ'(R), is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of R, and two distinct vertices x and y are adjacent if and only if x∉Ry and y∉Rx. In this paper, first we study the Laplacian spectrum of Γ'(Zn). We show that the graph Γ'(Zpq) is Laplacian integral. Further, we obtain the Laplacian spectrum of Γ'(Zn) for n=pn1qn2, where n1,n2∈N and p, q are distinct primes. In order to study the Laplacian spectral radius and algebraic connectivity of Γ'(Zn), we characterized the values of n for which the Laplacian spectral radius is equal to the order of Γ'(Zn). Moreover, the values of n for which the algebraic connectivity and vertex connectivity of Γ'(Zn) coincide are also described. At the final part of this paper, we obtain the Wiener index of Γ'(Zn) for arbitrary n.