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

Abstract

<abstract><p>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) \rightarrow \{0, 1, 2, ..., \}, $ such as $ |\varpi(\mu')-\varpi(\mu'')|\geq diam(G)+1-d(\mu', \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')-\varpi(\mu'')|: \mu', \mu'' \in V(G)\}. $ $ rn(G) = min_{\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 \geq 3) $ vertices, along with a complete bipartite graph $ K_{m, m} $ consisting of $ m $ v/ertices, where $ m = 2 $.</p></abstract>