Finite groups whose commuting graphs are line graphs

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Document Type : Original paper

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Department of Mathematics, Central University of Rajasthan, Ajmer, India

Abstract

The commuting graph Γ(G) of a group G is the simple undirected graph with group elements as a vertex set and two elements x and y are adjacent if and only if xy=yx in G. By eliminating the identity element of G and all the dominant vertices of Γ(G), the resulting subgraphs of Γ(G) are Γ∗(G) and Γ∗∗(G), respectively. In this paper, we classify all the finite groups G such that the graph Δ(G)∈{Γ(G),Γ∗(G),Γ∗∗(G)} is the line graph of some graph. We also classify all the finite groups G whose graph Δ(G)∈{Γ(G),Γ∗(G),Γ∗∗(G)} is the complement of line graph.

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References

[1] F. Ali, M. Salman, and S. Huang, On the commuting graph of dihedral group, Comm. Algebra 44 (2016), no. 6, 2389–2401.  https://doi.org/10.1080/00927872.2015.1053488
[2] Z. Barati, Line zero divisor graphs, J. Algebra Appl. 20 (2021), no. 9, Article ID:  2150154. https://doi.org/10.1142/S0219498821501541
[3] L.W. Beineke, Characterizations of derived graphs, J. Combin. Theory 9 (1970), no. 2, 129–135. https://doi.org/10.1016/S0021-9800(70)80019-9
[4] S. Bera, Line graph characterization of power graphs of finite nilpotent groups, Comm. Algebra 50 (2022), no. 11, 4652–4668.  https://doi.org/10.1080/00927872.2022.2069793
[5] R. Brauer and K.A. Fowler, On groups of even order, Ann. Math. 62 (1955), no. 3, 565–583.
[6] D. Bundy, The connectivity of commuting graphs, J. Comb. Theory Ser. A. 113 (2006), no. 6, 995–1007. https://doi.org/10.1016/j.jcta.2005.09.003
[7] R. Carleton and M.L. Lewis, The commuting graph of a solvable A-group, J. Group Theory 28 (2025), no. 1, 165–178. https://doi.org/10.1515/jgth-2023-0076
[8] G. Chartrand, L. Eroh, M.A. Johnson, and O.R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math. 105 (2000),  no. 1-3, 99–113.  https://doi.org/10.1016/S0166-218X(00)00198-0
[9] C. Clifton, Commutativity in non-abelian groups, Senior Project Report, Whitman College (2010).
[10] S. Malviy and V. Kakkar, Commuting graph of non-abelian groups of order p4 with center having p elements, Discrete Math. Algorithms Appl. (2024), Article ID: 2450080.  https://doi.org/10.1142/S1793830924500800
[11] M. Manisha, P. Parveen, and J. Kumar, Line graph characterization of the order supergraph of a finite group, Commun. Comb. Optim. (2024), In press.  https://doi.org/10.22049/cco.2024.29375.1962
[12] M. Mirzargar, P.P. Pach, and A.R. Ashrafi, Remarks on commuting graph of a finite group, Electron. Notes Discrete Math. 45 (2014), 103–106.  https://doi.org/10.1016/j.endm.2013.11.020
[13] P. Parveen and J. Kumar, On finite groups whose power graphs are line graphs, J. Algebra Appl. 24 (2024), no. 12, Article ID: 2550285.  https://doi.org/10.1142/S0219498825502858
[14] A. Sebő and E. Tannier, On metric generators of graphs, Math. Oper. Res. 29 (2004), no. 2, 383–393. https://doi.org/10.1287/moor.1030.0070
[15] M. Torktaz and A.R. Ashrafi, Commuting graph of CA- groups, Proyecciones 42 (2023), no. 1, 1–17. http://dx.doi.org/10.22199/issn.0717-6279-4488
[16] Mantel W., Problem 28, soln. by H. Gouventak, W. Mantel, J. Teixeira de Mattes, F. Schuh and W.A. Wythoff, Wiskundige Opgaven 10 (1907), 60–61.
[17] N. Yang and A.S. Mamontov, (2, 3)-generated groups with small element orders,  Algebra Logic 60 (2021), no. 3, 217–222.  https://doi.org/10.1007/s10469-021-09644-w
Communications in Combinatorics and Optimization

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Available Online from 26 May 2025

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