Let R be a commutative ring and be an ideal of R . By S. P. Redmond, the ideal-based zero-divisor graph ΓI (R) is defined by the following statement: the vertices set is a set {x∈R-I|xy∈ I for some y∈R-I } and the edges set is a set {[x, y]} where [x, y] is an edge such that xy ∈I for distinct two vertices x and y. Let R = Z18 when Z is the integers and (18) is an ideal of Z generated by 18. This graph is easily understood by students. The coefficients of a characteristic polynomial are related to triangles, 4 cycles and 2-matchings. We use the educational diagrams. These results are useful to students.