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Let be a finite, connected, undirected graph with graph diameter and graph distance between vertices and . A radio labeling of a graph is labeling using distinct nonnegative integers such that for every pair of distinct vertices , in the vertex set of . Then the radio number of , commonly denoted , is the smallest integer such that has a radio labeling with .

The radio number of path graphs and cycle graphs were determined by Liu and Zhu (2005). The following table summarizes some known results for a number of special families of graphs.

Graph Diameter, Graph Distance

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## References

Bantva, D. "Further Results on the Radio Number of Trees." 25 May 2018. https://arxiv.org/abs/1805.10083.Chartrand, G.; Erwin, D.; Harary, F.; and Zhang, P. "Radio Labelings of Graphs." Bull. Inst. Combin. Appl. 33, 77-85, 2001.Chartrand G.; and Zhang, P. "Radio Colorings of Graphs--A Survey." Int. J. Comput. Appl. Math. 2, 237-252, 2007.Griggs, J. R. and Yeh, R. K. "Labeling Graphs with Condition at Distance. 2." SIAM J. Disc. Math. 5, 586-595, 1992.Liu, D. "Radio Number for Trees." Disc. Math. 308, 1153-1164, 2008.Liu, D. D.-F.; Zhu, X. "Multilevel Distance Labelings for Paths and Cycles." SIAM J. Disc. Math. 19, 610-621, 2005.Zhang, P. "Radio Labeling of Cycles." Ars Combin. 65, 21-32, 2002.