Radio number of $2-$ super subdivision for path related graphs

We studied radio labelings of graphs in response to the Channel Assignment Problem (CAP). In a graph $G,$ the radio labeling is a mapping $\varpi :V(G)\to \{0,1,2,...,\},$ such as $|\varpi ({\mu }^{\prime })-\varpi ({\mu }^{″})|\ge diam(G)+1-d({\mu }^{\prime },{\mu }^{″}).$ The label of $\mu$ for under $\varpi$ is defined by the integer $\varpi (\mu ),$ and the span under is defined by $span(\varpi )=max\{|\varpi ({\mu }^{\prime })-\varpi ({\mu }^{″})|:{\mu }^{\prime },{\mu }^{″}\in V(G)\}.$ $rn(G)=mi{n}_{\varpi }span(\varpi )$ is defined as the radio number of $G$ when the minimum over all radio labeling $\varpi$ of $G$ is taken. $G$ is said to be optimal if its radio labeling is $span(\varpi )=rn(G).$ A graph H is said to be an $m$ super subdivision if $G$ is replaced by the complete bipartite graph $K_{m,m}$ with $m=2$ in such a way that the end vertices of the edge are merged with any two vertices of the same partite set $X$ or $Y$ of $K_{m,m}$ after removal of the edge of $G$. Up to this point, many lower and upper bounds of $rn(G)$ have been found for several kinds of graph families. This work presents a comprehensive analysis of the radio number $rn(G)$ for a graph $G$, with particular emphasis on the $m$ super subdivision of a path $P_n$ with $n(n\ge 3)$ vertices, along with a complete bipartite graph $K_{m,m}$ consisting of $m$ v/ertices, where $m=2$.