\(C_k\)-E-Super Magic Graceful Labeling of Some Families of Graphs

Abstract

If each edge in \(E(G)\) of a subgraph of \(G\) is isomorphic to \(H\), the graph \(G\) has an \(H\)-covering. A bijection \(f: V(G) \cup E(G) \rightarrow\{1,2, \ldots, p+q\}\) is \(H\) - \(E\)-super magic graceful labeling (H-E-SMGL) with the property \(f(E(G))=\{1,2, \ldots, q\}\) and for each subgraph \(H^{\prime}\) of \(G\) isomorphic to \(H, \sum_{v \in V\left(H^{\prime}\right)} f(v)-\sum_{e \in E\left(H^{\prime}\right)} f(e)=M\) for some positive integer \(M\).Applying the definition of \(H\)-E-SMGL, in this article we study \(C_k\)-E-super magic labeling of some families of graphs such as generalized fan graph and generalization of a graph attained at connecting of a star \(K_{1, n}\) with one isolated vertex.