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

For a simple graph G = (V, E), an L(2, 1)-labeling is an assignment of non-negative integer labels to vertices of G. An L(2, 1)-labeling of G must satisfy two conditions: adjacent vertices in 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 G, denoted by lambda(G), represents the smallest positive integer l for which an L(2, 1)-labeling exists, the vertices of G are provided labels from the set {0,1, ... , l}. 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 Z(pn), Z(pn )x Z(qm), and F(q )x Z(pn). This exploration leads us to compute the exact value of L(2, 1)-labeling number of these graphs.