Some Results on the Comaximal Colon Ideal Graph

In this paper, r is a commutative ring with a non-zero identity andm 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.