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ESTD Year: 2017 | Impact Factor (2026): 8.7
DOI Prefix: 10.47001/IRJIET
ESTD Year: 2017 | Impact Factor (2026): 8.7
DOI Prefix: 10.47001/IRJIET
Vol 10 No 6 (2026): Volume 10, Issue 6, June 2026 | Pages: 83-95
International Research Journal of Innovations in Engineering and Technology
OPEN ACCESS | Research Article | Published Date: 08-06-2026
Integer labeling of graphs the assignment of whole numbers to structural elements under prescribed sum conditions occupies a prominent position within discrete mathematics. When the labels assigned to edges form a consecutive sequence starting from one, and every vertex sees the same total across its incident edges, the configuration is called super magic in the sense introduced by Akka and Warad (2010). Despite a substantial body of work characterizing which graph families admit such labelings, the question of how to construct them inductively has remained largely unexplored.
This paper focuses on the generalized Petersen graph P(N, 1), a well-studied cubic graph family parameterized by an odd integer N ≥ 3. Three interrelated contributions are presented. First, the magic constant formula C = (19N + 3)/2 is derived through a direct algebraic argument from first principles. Second, the first induction-based proof of the super magic property of P(N, 1) is developed, identifying a vertex-insertion mechanism that transfers a valid labeling from P(2m−1, 1) to P(2m+1, 1) while increasing the magic constant by exactly 19 at each step. Third, computationally verified labelings for P(3, 1), P(5, 1), and P(7, 1) are presented with every edge sum checked explicitly. All diagrams were rendered programmatically using Python 3.11, NumPy 1.26, and Matplotlib 3.8; the inputs, layout logic, and generation procedure are documented fully in Section 7.
super magic labeling, generalized Petersen graph, magic constant, mathematical induction, vertex-insertion construction, cubic graphs, Python, Matplotlib, computational graph theory.
Mallikarjun Ghaleppa, Amit Kumar Yadav, Annadurai Manickam, Amabelle Oliva Enanoria, & Ramesh Palanisamy. (2026). Super Magic Labeling of Generalized Petersen Graphs P(N, 1): An Induction-Based Proof with Computational Verification. International Research Journal of Innovations in Engineering and Technology - IRJIET, 10(6), 83-95. Article DOI https://doi.org/10.47001/IRJIET/2026.106009
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