On the cozero-divisor graphs associated to rings

Abstract

AbstractLet R be a ring with unity. The cozero-divisor graph of a ring R, denoted by [Formula: see text] 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 [Formula: see text] and [Formula: see text] In this paper, first we study the Laplacian spectrum of [Formula: see text] We show that the graph [Formula: see text] is Laplacian integral. Further, we obtain the Laplacian spectrum of [Formula: see text] for [Formula: see text] where [Formula: see text] and p, q are distinct primes. In order to study the Laplacian spectral radius and algebraic connectivity of [Formula: see text] we characterized the values of n for which the Laplacian spectral radius is equal to the order of [Formula: see text] Moreover, the values of n for which the algebraic connectivity and vertex connectivity of [Formula: see text] coincide are also described. At the final part of this paper, we obtain the Wiener index of [Formula: see text] for arbitrary n.

Keywords

• Cozero-divisor graph
• ring of integers modulo n
• Laplacian spectrum
• Wiener index
• 05C25
• 05C50