Computing edge weights of magic labeling on rooted products of graphs
Summary: Labeling of graphs with numbers is being explored nowadays due to its diverse range of applications in the fields of civil, software, electrical, and network engineering. For example, in network engineering, any systems interconnected in a network can be converted into a graph and specific numeric labels assigned to the converted graph under certain rules help us in the regulation of data traffic, connectivity, and bandwidth as well as in coding/decoding of signals. Especially, both antimagic and magic graphs serve as models for surveillance or security systems in urban planning. In [SUT J. Math. 34, No. 2, 105--109 (1998; Zbl 0918.05090)], \textit{H. Enomoto} et al. introduced the notion of super \(\left(a, 0\right)\) edge-antimagic labeling of graphs. In this article, we shall compute super \(\left(a, 0\right)\) edge-antimagic labeling of the rooted product of \(P_n\) and the complete bipartite graph \(\left( K_{2, m}\right)\) combined with the union of path, copies of paths, and the star. We shall also compute a super \(\left(a, 0\right)\) edge-antimagic labeling of rooted product of \(P_n\) with a special type of pancyclic graphs. The labeling provided here will also serve as super \(\left( a^\prime, 2\right)\) edge-antimagic labeling of the aforesaid graphs. All the structures discussed in this article are planar. Moreover, our findings have also been illustrated with examples and summarized in the form of a table and \(3D\) plots.
scientific article; zbMATH DE number 7260951
scientific article