Abstract
For a group G, the generating graph \({\Gamma }(G)\) is defined as the graph with the vertex set G, and any two distinct vertices of \({\Gamma }(G)\) are adjacent if they generate G. In this paper, we study the generating graph of \(D_n,\) where \(D_n\) is a Dihedral group of order 2n. We explore various graph theoretic properties, and determine complete spectrum of the adjacency and the Laplacian matrix of \({\Gamma }(D_n)\). Moreover, we compute some distance and degree based topological indices of \({\Gamma }(D_n)\).
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References
Ahmadi, M.R., Jahani-Nezhad, R.: Energy and wiener index of zero-divisor graphs. Iranian. J. Math. Chem. 2, 45–51 (2011)
Barik, S., Kalita, D., Pati, S., Sahoo, G.: Spectra of graphs resulting from various graph operations and products: a survey. Spec. Matrices 6, 323–342 (2018)
CONRAD, K.: Subgroup series ii. https://kconrad.math.uconn.edu/blurbs/grouptheory/subgpseries2.pdf
Crestani, E., Lucchini, A.: The generating graph of finite soluble groups. Isr. J. Math. 198, 63–74 (2013)
Detomi, E., Lucchini, A.: Crowns and factorization of the probabilistic zeta function of a finite group. J. Algebra 265, 651–668 (2003)
Godsil, C., Royle, G.F.: Algebraic Graph Theory. Springer, Berlin (2001)
Guralnick, R., Kantor, W.: Probabilistic generation of finite simple groups. J. Algebra 234, 743–792 (2000)
Gutman, I.: Selected properties of the Schultz molecular topological index. J. Chem. Inf. Comput. Sci. 34, 1087–1089 (1994)
Gutman, I., Trinajstić, N.: Graph theory and molecular orbitals. Total \(\varphi \)-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17, 535–538 (1972)
Klein, D.J., Lukovits, I., Gutman, I.: On the definition of the hyper-wiener index for cycle-containing structures. J. Chem. Inf. Comput. Sci. 35, 50–52 (1995)
Liebeck, M.W., Shalev, A.: Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky. J. Algebra 184, 31–57 (1996)
Lucchini, A.: The diameter of the generating graph of a finite soluble group. J. Algebra 492, 28–43 (2017)
Lucchini, A.: Finite groups with planar generating graph. Australas. J. Combin. 76, 220–225 (2020)
Lucchini, A., Maróti, A.: On the clique number of the generating graph of a finite group. Proc. Am. Math. Soc. 137, 3207–3217 (2009)
Lucchini, A., Maróti, A.: Some results and questions related to the generating graph of a finite group. In: Ischia Group Theory 2008, World Sci. Publ., Hackensack, 183–208 (2009)
Mirzargar, M., Ashrafi, A.R.: Some distance-based topological indices of a non-commuting graph. Hacet. J. Math. Stat. 41, 515–526 (2012)
Sarmin, N.H., Alimon, N.I., Erfanian, A.: Topological indices of the non-commuting graph for generalised quaternion group. Bull. Malays. Math. Sci. Soc. 43, 3361–3367 (2020)
H. P. Schultz, Topological organic chemistry. 1. graph theory and topological indices of alkanes, Journal of Chemical Information and Computer Sciences, 29, 227–228 (1989)
West, D.B., et al.: Introduction to Graph Theory, vol. 2. Prentice Hall, Upper Saddle River (2001)
Wiener, H.: Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)
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The authors express their sincere gratitude to the learned referee for her/his meticulous reading and valuable suggestions which have definitely improved the quality of the article.
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The second named author is supported by Shiv Nadar Institution of Eminence Ph.D. Fellowship.
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Reddy, A.S., Samant, K. Generating Graphs of Finite Dihedral Groups. Results Math 78, 200 (2023). https://doi.org/10.1007/s00025-023-01963-x
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DOI: https://doi.org/10.1007/s00025-023-01963-x