l(2,1)-labeling of some zero-divisor graphs associated with commutative rings

Let $mathcal g = (mathcal v, mathcal e)$ be a simple graph, an $l(2,1)$-labeling of $mathcal g$ is an assignment of labels from non-negative integers to vertices of $mathcal g$ such that adjacent vertices get labels which differ by at least by two, and vertices which are at distance two from each other get different labels. the $lambda$-number of $mathcal g$, denoted by $lambda(mathcal g)$, is the smallest positive integer $ell$ such that $mathcal g$ has an $l(2,1)$-labeling with all labels as  members of the set ${ 0, 1, dots, ell }$. the zero-divisor graph of a finite commutative ring $r$ with unity, denoted by $gamma(r)$, is the simple graph whose vertices are all zero divisors of $r$ in which two vertices $u$ and $v$ are adjacent  if and only if $uv = 0$ in $r$. in this paper, we investigate $l(2,1)$-labeling of some  zero-divisor graphs. we study the textit{partite truncation}, a graph operation that allows us to obtain a reduced graph of relatively small order from a graph of significantly larger order. we establish the relation between  $lambda$-numbers of the graph  and its partite truncated one. we make use of the operation textit{partite truncation} to contract the zero-divisor graph of a reduced ring to the zero-divisor graph of a boolean ring.