Some Results on the Comaximal Colon Ideal Graph

In this paper, R is a commutative ring with a non-zero identity and M is a unital R-module. We introduce the comaximal colon ideal graph C ∗ (R) and colon submodule graph C ∗ (M); and study the interplay between the graph-theoretic properties and the corresponding algebraic structures. C ∗ (R) is a simple connected supergraph of the comaximal ideal graph C(R) with diam(C ∗ (R)) ≤ 2. Moreover, we prove that if |V(C ∗ (R)| ≥ 3, then gr(C ∗ (R)) = 3. We prove that if |Max(R)| = n, then C ∗ (R) containing a complete n-partite subgraph. Also if M is a finitely generated multiplication module, then C ∗ (R) ∼= C ∗ (M). Moreover, for Z-module Zn which n is not a prime, C ∗ (Zn) ∼= Kd(n), where d(n) is the number of all divisors of the positive integer n other than 1 and n.