Universal adjacency spectrum of the cozero-divisor graph and its complement on a finite commutative ring with unity

For a finite simple undirected graph G, the universal adjacency matrix U(G) is a linear combination of the adjacency matrix A(G), the degree diagonal matrix D(G), the identity matrix I and the all-ones matrix J, that is U(G)=alpha A(G)+beta D(G)+gamma I+eta J, where alpha,beta,gamma,eta is an element of R and alpha not equal 0. The cozero-divisor graph Gamma '(R) of a finite commutative ring R with unity is a simple undirected graph with the set of all nonzero nonunits of R as vertices and two vertices x and y are adjacent if and only if x is not an element of y R and y is not an element of x R. In this paper, we study structural properties of Gamma '(R) by defining an equivalence relation on its vertex set in terms of principal ideals of the ring R. Then we obtain the universal adjacency eigenpairs of Gamma '(R) and its complement, and as a consequence one may obtain several spectra like the adjacency, Seidel, Laplacian, signless Laplacian, normalized Laplacian, generalized adjacency and convex linear combination of the adjacency and degree diagonal matrix of Gamma '(R) and Gamma '(R)((sic))in an unified way. Moreover, we get the universal adjacency eigenpairs of the cozero-divisor graph and its complement for a reduced ring and the ring of integers modulo m in a simpler form.