On the cozero-divisor graphs associated to rings. Issue 3 (2nd September 2022) - Mathil, Praveen
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 ∉ R y and y ∉ R x . In this paper, first we study the Laplacian spectrum of Γ ′ ( Z n ) . We show that the graph Γ ′ ( Z p q ) is Laplacian integral. Further, we obtain the Laplacian spectrum of Γ ′ ( Z n ) for n = p n 1 q n 2, where n 1, n 2 ∈ N and p, q are distinct primes. In order to study the Laplacian spectral radius and algebraic connectivity of Γ ′ ( Z n ), we characterized the values of n for which the Laplacian spectral radius is equal to the order of Γ ′ ( Z n ) . Moreover, the values of n for which the algebraic connectivity and vertex connectivity of Γ ′ ( Z n ) coincide are also described. At the final part of this paper, we obtain the Wiener index of Γ ′ ( Z n ) for arbitrary n .