Universal adjacency spectrum of the looped zero divisor graph for a finite commutative ring with unity.

For a finite undirected looped graph G ˚ , the universal adjacency matrix U (G ˚) is a linear combination of the adjacency matrix A (G ˚) , the degree matrix D (G ˚) , the identity matrix I and the all-ones matrix J , that is U (G ˚) = α A (G ˚) + β D (G ˚) + γ I + η J , where α , β , γ , η ∈ ℝ and α ≠ 0. For a finite commutative ring R with unity, the looped zero divisor graph Γ ˚ (R) is an undirected graph with the set of all nonzero zero divisors of R as vertices and two vertices (not necessarily distinct) x and y are adjacent if and only if x y = 0. In this paper, we study some structural properties of Γ ˚ (R) by defining an equivalence relation on its vertex set. Then we obtain the universal adjacency eigenpairs of Γ ˚ (R) , and as a consequence several spectra like the adjacency, Seidel, Laplacian, signless Laplacian, normalized Laplacian, generalized adjacency and convex linear combination of the adjacency and degree matrix of Γ ˚ (R) can be obtained in a unified way. Moreover, we get the structural properties and the universal adjacency eigenpairs of the looped zero divisor graph of a reduced ring in a simpler form. [ABSTRACT FROM AUTHOR]