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Abstract
Generalized quarternion group (Q_(4n)) is a group of order $4n$ that is generated by two elements x and y with the properties x^{2n}=y^4=e and xy=yx^{-1}. The coprime graph of Q_{4n}, denoted by Omega_{Q_{4n}}, is a graph with the vertices are elements of Q_{4n} and the edges are generated by two elements that have coprime order. The first result of this paper presented that Omega_{Q_{4n}} is a tripartite graph for n is odd and Omega_{Q_{4n}} is a star graph for n is even. The second one presented the connectivity indices of Omega_{Q_{4n}}. Connectivity indices of a graph is a research area in mathematics that popularly applied in chemistry. There are six indices that are presented in this paper, those are first Zagreb index, second Zagreb index, Wiener index, hyper-Wiener index, Harary index, and Szeged index.
Generalized quaternion group (Q4n) is a group of order 4n that is generated by two elements x and y with the properties x 2n = y 4 = e and xy = yx−1 . The coprime graph of Q4n, denoted by ΩQ4n , is a graph with the vertices are elements of Q4n and the edges are formed by two elements that have coprime order. The first result of this paper presents that ΩQ4n is a tripartite graph for n is an odd prime and ΩQ4n is a star graph for n is a power of 2. The second one presents the connectivity indices of ΩQ4n . Connectivity indices of a graph is a research area in mathematics that popularly applied in chemistry. There are six indices that are presented in this paper, those are first Zagreb index, second Zagreb index, Wiener index, hyper-Wiener index, Harary index, and Szeged index.Keywords
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References
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References
Alimon, N.I., Sarmin, N.H., and Erfanian, A., Topological indices of non-commuting graph of dihedral groups, Malaysian J. Fundam. Appl. Sci., 14 (2018), 473-6.
Chartrand, G., Lesniak, L., and Zhang, P., Graphs and Digraphs, Chapman and Hall/CRC, 2015.
Das, K.C., Xu, K., and Nam, J., ”Zagreb indices of graphs”, Frontiers of Mathematics in China, 10(3) (2015), 567582.
Dobrynin, A.A., Entringer, R., and Gutman, I., ”Wiener index of trees: theory and applications”, Acta Applicandae Mathematica, 66(3) (2001), 211-249.
Fraleigh, J.B., A first course in abstract algebra, Pearson Education India, 2003.
Klavzar, S., Rajapakse, A., and Gutman, I., ”The szeged and the wiener index of graphs”, Applied Mathematics Letters, 9(5) (1996), 45-49.
MA, X., WEI, H., and YANG, L. ”The coprime graph of a group”, International Journal of Group Theory, (2014),
Sarmin, N.H., Alimon, N.I., and Erfanian, A., Topological indices of the non-commuting graph for generalised quaternion group, Bulletin of the Malaysian Mathematical Sciences Society, (2019), 1-7.
Wiener. H., ”Structural determination of paraffin boiling points”, Journal of the American chemical society, 69(1) (1947), 17-20.
Xu, K., and Das, K.C., ”On harary index of graphs”, Discrete applied mathematics, 159(15) (2011), 1631-1640.
Yamasaki, Y., Ramanujan cayley graphs of the generalized quaternion groups and the hardy littlewood conjecture In Mathematical modelling for next-generation cryptography, Springer,