Non-reduced rings of small order and their maximal graph

Let R
be a commutative ring with nonzero identity. Let Γ(R) denotes the maximal graph corresponding to the non-unit elements of R, that is, Γ(R)
is a graph with vertices the non-unit elements of R, where two distinct
vertices a and b are adjacent if and only if there is a maximal ideal of R
containing both. In this paper, we investigate that for a given positive integer n, is there a non-reduced ring R with n non-units? For n≤100, a complete list of non-reduced decomposable rings R=∏ki=1Ri (up to cardinalities of constituent local rings Ri's) with n non-units is given. We also show that for which n, (1≤n≤7500), |Center(Γ(R))| attains the bounds in the inequality 1≤|Center(Γ(R))|≤n and for which n, (2≤n≤100), |Center(Γ(R))|
attains the value between the bounds