H-E-Super magic graceful labelings of graphs: H-E-SMGL - Duraisamy, Kumar; M. Elakkiya,; J. Sebastian Lawrance,; M. Bhuvaneshwari, (Communications in Mathematics and Applications, 2025)
Let $G$ be a simple graph with $p$ vertices and $q$ edges. A simple graph $G$ admits an $H$-covering in $E(G)$ belongs to a subgraph of $G$ isomorphic to $H$. The graph $G$ is said to be $H$-magic if there exists an one one onto $μ$ from $V(G) \cup E(G)$ onto $→ \{ 1,2 , …, p + q \}$ such that for every subgraph $H^{'}$ of $G$ isomorphic to $H$, $ \sum\limits_{v \in V(H^{'})} μ(v) + \sum\limits_{e \in E(H^{'})} μ(e) = M$ for some positive integer $M$. An $H$-$E$-super magic graceful labeling ($H$-$E$-SMGL) is an one one onto $μ$ from $V(G) \cup E(G)$ onto $→ \{ 1,2 , …, p + q \}$ with the condition that $μ(E(G)) = \{ 1,2 , …, q\}$ such that, $ \sum\limits_{v \in V(H^{'})} μ(v) - \sum\limits_{e \in E(H^{'})} μ(e) = M$. In this paper, we study $H$-$E$-SMGL of fans, graphs obtained by joining a star $K_{1,n}$ with one isolated vertex, books and grids.