Signless Laplacian spectrum of the cozero-divisor graph of the commutative ring ℤ_𝑛

Abstract

Let 𝑅 be a commutative ring with identity 1 ≠ 0 and let Z ⁢ ( R ) ′ be the set of all non-zero and non-unit elements of ring 𝑅. Further, Γ ′ ⁢ ( R ) denotes the cozero-divisor graph of 𝑅, is an undirected graph with vertex set Z ⁢ ( R ) ′ , and w ∉ z ⁢ R and z ∉ w ⁢ R if and only if two distinct vertices 𝑤 and 𝑧 are adjacent, where q ⁢ R is the ideal generated by the element 𝑞 in 𝑅. In this paper, we find the signless Laplacian eigenvalues of the graphs Γ ′ ⁢ ( Z n ) for n = p 1 N ⁢ p 2 ⁢ p 3 and p 1 N ⁢ p 2 M ⁢ p 3 , where p 1 , p 2 , p 3 are distinct primes and N , M are positive integers. We also show that the cozero-divisor graph Γ ′ ⁢ ( Z p 1 ⁢ p 2 ) is a signless Laplacian integral.