On L(2, 1)-labeling of zero-divisor graphs of finite commutative rings

Abstract

For a simple graph $\mathcal {G}= (\mathcal {V}, \mathcal {E})$, an L(2, 1)-labeling is an assignment of non-negative integer labels to vertices of $\mathcal {G}$. An L(2, 1)-labeling of $\mathcal {G}$ must satisfy two conditions: adjacent vertices in $\mathcal {G}$ should get labels which differ by at least two, and vertices at a distance of two from each other should get distinct labels. The $\lambda$-number of $\mathcal {G}$, denoted by $\lambda (\mathcal {G})$, represents the smallest positive integer $\ell$ for which an L(2, 1)-labeling exists, the vertices of $\mathcal {G}$ are provided labels from the set $\{0, 1, \dots , \ell \}$. Let $\Gamma (R)$ be a zero-divisor graph of a finite commutative ring R with unity. In $\Gamma (R)$, vertices represent zero-divisors of R, and two vertices x and y are adjacent if and only if $xy = 0$ in R. The methodology of the research involves a detailed investigation into the structural aspects of zero-divisor graphs associated with specific classes of local and mixed rings, such as $\mathbb {Z}_{p^n}$, $\mathbb {Z}_{p^n} \times \mathbb {Z}_{q^m}$, and $\mathbb {F}_{q}\times \mathbb {Z}_{p^n}$. This exploration leads us to compute the exact value of L(2, 1)-labeling number of these graphs.