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Radio Labeling of Strong Prismatic Network With Star

The rapid development of wireless communication has made efficient spectrum assignment a crucial factor in enhancing network performance. As a combinatorial optimization model for channel assignment, the radio labeling is recognized as an NP-hard problem. Therefore, converting the spectrum assignment problem into the radio labeling of graphs and studying the radio labeling of specific graph classes is of great significance. For $G$, a radio labeling $φ: V(G) \to \{0, 1, 2, \ldots\}$ is required to satisfy $|φ(u) - φ(v)| \geq \text{diam}(G) + 1 -d_G(u, v)$, where ${diam(G)}$ and $d_G(u, v)$ are diameter and distance between $u$ and $v$. For a radio labeling $φ$, its $\text{span}$ is defined as the largest integer assigned by $φ$ to the vertices of $G$; the radio labeling specifically denotes the labeling with the minimal span among possible radio labeling. The strong product is a crucial tool for constructing regular networks, and studying its radio labeling is necessary for the design of optimal channel assignment in wireless networks. Within this manuscript, we discuss the radio labeling of strong prismatic network with star, present the relevant theorems and examples, and propose a parallel algorithm to improve computational efficiency in large-scale network scenarios.