On the cozero-divisor graphs associated to rings

Let R be a ring with unity. The cozero-divisor graph of a ring R, denoted by Gamma '(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 is not an element of Ry and y is not an element of Rx. In this paper, first we study the Laplacian spectrum of Gamma '(Z(n)). We show that the graph Gamma '(Z(pq)) is Laplacian integral. Further, we obtain the Laplacian spectrum of Gamma '(Z(n)) for n = p(n1)q(n2), where n1,n2 is an element of N and p, q are distinct primes. In order to study the Laplacian spectral radius and algebraic connectivity of Gamma '(Z(n)), we characterized the values of n for which the Laplacian spectral radius is equal to the order of Gamma '(Z(n)). Moreover, the values of n for which the algebraic connectivity and vertex connectivity of Gamma '(Z(n)) coincide are also described. At the final part of this paper, we obtain the Wiener index of Gamma '(Z(n)) for arbitrary n.