Independent domination polynomial for the cozero divisor graph of the ring of integers modulo n

Abstract

The cozero divisor graph Γ (cid:48) ( R ) of a commutative ring R is a simple graph whose vertex set is the set of non-zero non-unit elements of R such that two distinct vertices x and y of Γ (cid:48) ( R ) are adjacent if and only if x / ∈ Ry and y / ∈ Rx , where Rx is the ideal generated by x . In this article, the independent domination polynomial of Γ (cid:48) ( Z n ) is found for n ∈ { p 1 p 2 , p 1 p 2 p 3 , p n 1 1 p 2 } , where p i ’s are primes, n 1 is an integer greater than 1 , and Z n is the integer modulo ring. It is shown that the independent domination polynomial of Γ (cid:48) ( Z p 1 p 2 ) has only one real root. It is also proved that these polynomials are not unimodal but are log-concave under certain conditions.