Ideal-based quasi cozero divisor graph of a commutative ring

Let R be a commutative ring with identity, and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by ΓI(R), is the graph whose vertices are the set ∈ R I | xy ∈ I for some y ∈ R I\, where distinct vertices x and y are adjacent if and only if xy ∈ I. The cozero-divisor graph with respect to I, denoted by Γ''I(R), is the graph of R with vertices ∈ R I | xR + I ≠ R\, and two distinct vertices x and y are adjacent if and only if x ∉ yR + I and y ∉ xR + I. In this paper, we introduce and investigate an undirected graph QΓ''I(R) of R with vertices ∈ R √I | xR + I ≠ R and xR + √I = xR + I\ and two distinct vertices x and y are adjacent if and only if x ∉ yR + I and y ∉ xR + I.