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

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