ON SUPER a; d -EAT VALUATION OF SUBDIVIDED CATERPILLAR

Yıl 2019, Cilt: 9 Sayı: 4, 693 - 703, 01.12.2019

A. Raheem M. Javaid M. A. Umar G. C. Lau

Öz

Let G = V G , E G be a graph with v = |V G | vertices and e = |E G | edges. A bijective function λ : V G ∪ E G ↔ {1, 2, . . . , v + e} is called an a, d - edge antimagic total EAT labeling valuation if the weight of all the edges {w xy : xy ∈ E G } form an arithmetic sequence starting with first term a and having common difference d, where w xy = λ x + λ y + λ xy . And, if λ V = {1, 2, . . . , v} then G is super a, d -edge antimagic total EAT graph. In this paper, we determine the super a,d -edge antimagic total EAT labeling of the subdivided caterpillar for different values of the parameter d.

Anahtar Kelimeler

caterpillar, subdivided caterpillar, super a, d -EAT graph.

Kaynakça

• Ahmad, A., Baig, A. Q., and Imran, M., On super edge-magicness of graphs, Utilitas Math. to appear.
• Baˇca, M., Dafik, Miller, M., and Ryan, J., (2008), Edge-antimagic total labeling of disjoint unions of caterpillars, J. Comb. Math. Comb. Computing, 65, pp. 61–70.
• Baˇca, M., Y. Lin, Y., Miller, M., and Simanjuntak, R., (2001), New constructions of magic and antimagic graph labelings, Utilitas Math., 60, pp. 229–239.
• Baˇca, M., Lin, Y. and Muntaner-Batle, F.A., (2007), Super edge-antimagic labelings of the path-like trees, Utilitas Math.,73, pp. 117–128.
• Baˇca, M., and M. Miller, M., (2008), Super Edge-Antimagic Graphs, Brown Walker Press, Boca Raton, Florida USA.
• Baˇca, M.,, Lin, Y., Miller, M., and M. Z. Youssel, M. Z., Edge-antimagic graphs, Discrete Math., to appear.
• Baskoro, E. T., and Cholily. Y., (2004), Expanding super edge-magic graphs, Proc. ITB Sains and Tek.,36:2, pp. 117–125.
• Enomoto, H., Llado, A. S., Nakamigawa, T., and Ringle, G., (1980), Super edge-magic graphs, SUT J. Math., 34, pp. 105–109.
• Figueroa-Centeno, R. M., Ichishima, R., and Muntaner-Batle, F. A., (2001), The place of super edge- magic labeling among other classes of labeling, Discrete Math., 231, pp. 153–168.
• Figueroa-Centeno, R. M., Ichishima, R., and Muntaner-Batle, F. A., (2002), On super edge-magic graphs, Ars Combin., 64, pp. 81–95.
• Figueroa-Centeno, R. M., Ichishima, R., and Mantaner-Batle, F. A., (2005), On edge-magic labeling of certain disjoint union graphs, Australas. J. Combin., 32, pp. 225–242.
• Fukuchi, Y., A recursive theorem for super edge-magic labeling of trees, (2000), SUT J. Math., 36, pp. 279–285.
• J. A. Gallian, (2009), A dynamic survey of graph labeling, Electronic J. Combin.
• Hussain, M., Baskoro, E. T., Slamin, (2009) On super edge-magic total labeling of banana trees, Utilitas Math., 79, pp. 243–251.
• Javaid, M., (2014), On super edge-antimagic total labeling of generalized extended w-trees, Interna- tional Journl of Mathematics and soft Computing, 4, pp. 17–25.
• A. Kotzig, A., and Rosa, A., (1970), Magic valuations of finite graphs, Canad. Math. Bull., 13, pp. 451–461.
• Kotzig, A., and Rosa, A Magic valuation of complete graphs, (1972), Centre de Recherches Mathe- matiques, Universite de Montreal, CRM–175.
• Lee, S. M., and Shah, Q. X., All trees with at most 17 vertices are super edge-magic, (2002), 16th MCCCC Conference, Carbondale, University Southern Illinois.
• Raheem, A., Javaid, M., Baig, A. Q., (2016), Antimagic labeling of the union of subdivided stars, TWMS J. App. Eng. Math.,6(2), pp.244–250.
• Raheem, A., Javaid, M., Baig, A. Q., (2016), On antimagicness of generalized extene w-trees, science Internation Journal, 28(5),pp. 5057–5022.
• Raheem, A., (2018), On super (a, d)-edge antimagic total labeling of a subdivided stars, Ars Combin., 136, pp. 169–179.
• K. A. Sugeng. K. A., Miller, M., Baca, M., (a, d)-edge-antimagic total labeling of caterpillars, (2005), Lecture Notes Comput. Sci., 3330 pp. 169–180.
• West, D. B.,(1996), An Introduction to Graph Theory, Prentice-Hall.