Moreover, $G$ is said to be super edge-magic if $\phi(V(G))=\{1,2,\dots,|V(G)|\}.$ The super edge-magic deficiency of a graph $G$, denoted by $\mu_s(G)$, is the minimum nonnegative integer $n$ such that $G\cup nK_1,$ has a super edge-magic labeling, if such integer does not exist we define $\mu_s(G)$ to be $+\infty.$ In this paper, we study the super edge-magic deficiency of some Toeplitz graphs.
📌 3 Key Takeaways
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Objective & Hypothesis:A graph $G$ is called edge-magic if there exists a bijective function $\phi:V(G)\cup E(G)\to\{1, 2,\dots,|V(G)|+|E(G)|\}$ such that $\phi(x)+\phi(xy)+\phi(y)=c(\phi)$ is a constant for every edge $xy\in E(G)$, called the valence of $\phi$.
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Key Findings & Data:A graph $G$ is called edge-magic if there exists a bijective function $\phi:V(G)\cup E(G)\to\{1, 2,\dots,|V(G)|+|E(G)|\}$ such that $\phi(x)+\phi(xy)+\phi(y)=c(\phi)$ is a constant for every edge $xy\in E(G)$, called the valence of $\phi$.
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Conclusion & Impact:Moreover, $G$ is said to be super edge-magic if $\phi(V(G))=\{1,2,\dots,|V(G)|\}.$ The super edge-magic deficiency of a graph $G$, denoted by $\mu_s(G)$, is the minimum nonnegative integer $n$ such that $G\cup nK_1,$ has a super edge-magic labeling, if such integer does not exist we define $\mu_s(G)$ to be $+\infty.$ In this paper, we study the super edge-magic deficiency of some Toeplitz graphs.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:edge-magic · super edge-magic · Toeplitz graphs
Abstract
A graph $G$ is called edge-magic if there exists a bijective function $\phi:V(G)\cup E(G)\to\{1, 2,\dots,|V(G)|+|E(G)|\}$ such that $\phi(x)+\phi(xy)+\phi(y)=c(\phi)$ is a constant for every edge $xy\in E(G)$, called the valence of $\phi$. Moreover, $G$ is said to be super edge-magic if $\phi(V(G))=\{1,2,\dots,|V(G)|\}.$ The super edge-magic deficiency of a graph $G$, denoted by $\mu_s(G)$, is the minimum nonnegative integer $n$ such that $G\cup nK_1,$ has a super edge-magic labeling, if such integer does not exist we define $\mu_s(G)$ to be $+\infty.$ In this paper, we study the super edge-magic deficiency of some Toeplitz graphs.
GUPTA, A., NADEEM, M. F., AHMAD, A. (2022). On super edge-magic deficiency of certain Toeplitz graphs. Hacettepe Journal of Mathematics and Statistics, Cilt: 47 - Sayı: 3, 513–519. https://ajindex.com/articles/530185
B. D. Acharya, S. M. Hegde, Strongly indexable graphs, Discrete Math., 93 (1991), 123129. A. Ahmad, I. Javaid, M. F. Nadeem and R. Hasni, On Super edge-magic de ciency of some families related to ladder graphs, Australas. J. Combin., 51, 201208, 2011.
A. Ahmad, M. Baca, M. F. Nadeem, On edge irregularity strength of Toeplitz graphs, U.P.B. Sci. Bull., Series A, 78(4), 155162, 2016.
M. Baca, Y. Bashir, M. F. Nadeem, A. Shabbir, On super edge-antimagic total labeling of Toeplitz graphs, Springer Proceedings in Mathematics & Statistics, 98 (2015), 1-11.
H. Enomoto, A. Lladó, T. Nakamigawa, and G. Ringel, Super edge-magic graphs, SUT J. Math., 34, 105109, 1998.
R. M. Figueroa-Centeno, R. Ichishima and F.A. Muntaner-Batle, The place of super edge- magic labeling among other classes of labeling, Discrete Math., 231, 153168, 2001.
R. M. Figueroa-Centeno, R. Ichishima and F.A. Muntaner-Batle, On the super edge-magic de ciency of graphs, Electron. Notes Discrete Math., 11, 2002.
R. M. Figueroa-Centeno, R. Ichishima and F.A. Muntaner-Batle, On the super edge-magic de ciency of graphs, Ars Combin., 78, 2006.
A. Kotzig and A. Rosa, Magic valuaton of nite graphs, Canad. Math. Bull., 13(4), 451461, 1970.
A. A. G. Ngurah, R. Simanjuntak and E.T. Baskoro, On the super edge-magic de ciencies of graphs, Australas. J. Combin., 40, 314, 2008.
J.Y. Park, J.H. Choi and J.H. Bae, On super edge-magic labeling of some graphs, Bull. Korean Math. Soc., 45, No.1, 1121, 2008.