Lambda number of the reduced power graph of finite groups

The reduced power graph ${\mathrm{\Gamma }}_{RP}(G)$ of a group $G$ is a simple graph with vertex set $G$, where two distinct vertices $u$ and $v$ are adjacent if and only if $⟨u⟩\subset ⟨v⟩$ or $⟨v⟩\subset ⟨u⟩$. An $L(2,1)$-labeling of a graph $\mathrm{\Gamma }$ is a mapping from the vertex set of $\mathrm{\Gamma }$ to the set of non-negative integers such that the labels of adjacent vertices differ by at least $2$ and the labels of vertices at distance two are distinct. The lambda number of $\mathrm{\Gamma }$, denoted by $\lambda (\mathrm{\Gamma })$, is defined as the minimum possible span of labels among all $L(2,1)$-labelings of $\mathrm{\Gamma }$. In this paper, we establish upper and lower bounds for $\lambda ({\mathrm{\Gamma }}_{RP}(G))$ and characterize the conditions under which these bounds are achieved. As an application, we determine the exact value of $\lambda ({\mathrm{\Gamma }}_{RP}(G))$ for certain classes of finite groups.