University of GuilanJournal of Algebra and Related Topics2345-39316120180601Non-reduced rings of small order and their maximal graph3544302510.22124/jart.2018.10130.1097ENA.SharmaDepartment of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, IndiaA.GaurDepartment of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, IndiaJournal Article20180417Let $R$ be a commutative ring with nonzero identity. Let $Gamma(R)$ denotes the maximal graph corresponding to the non-unit elements of R, that is, $Gamma(R)$<br />is a graph with vertices the non-unit elements of $R$, where two distinct<br />vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $R$<br />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 $n leq 100$, a complete list of non-reduced decomposable rings $R = prod_{i=1}^{k}R_i$ (up to cardinalities of constituent local rings $R_i$'s) with n non-units is given. We also show that for which $n$, $(1leq n leq 7500)$, $|Center(Gamma(R))|$ attains the bounds in the inequality $1leq |Center(Gamma(R))|leq n$ and for which $n$, $(2leq nleq 100)$, $|Center(Gamma(R))|$ attains the value between the boundshttps://jart.guilan.ac.ir/article_3025_8bced734856b35561fe82af4e15d0d5c.pdf