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Some properties of various graphs associated with finite groups. (English) Zbl 1485.05075

In recent times, there are several graph constructions from algebraic structures. In particular, the study of graphs from groups right from the Cayley graph from finite groups attracted many researchers. Associating a graph with a group gives a tools of deriving properties of graphs through properties of the underlying groups and vice-versa. In recent times, there are several other well studied graphs from groups. Some of them to mention are power graph, commuting graph and super power graph of groups. Let \(G\) be a finite group and \(X\) a nonempty subset of \(G.\) The (undirected) power graph \(P(G,X),\) has \(X\) as its vertex set with two distinct elements of \(X\) joined by an edge when one is a power of the other. The directed power graph \(\overrightarrow{P}(G,X)\) is a directed graph with the set \(X\) of vertices and with all edges \((x, y)\) such that \(x\neq y\) and \(y\) is a power of \(x.\) The commuting graph \(C(G,X),\) has \(X\) as its vertex set with two distinct elements of \(X\) joined by an edge when they commute in \(G.\) In this paper, the authors are interested in some properties of the power graph and commuting graph associated with a finite group, using their tree-numbers. Among other things, it is shown that the simple group \(L_2(7)\) can be characterized through the tree-number of its power graph. Moreover, the classification of groups with power-free decomposition is presented. Finally, the authors obtain an explicit formula concerning the tree-number of commuting graphs associated with Suzuki simple groups.

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D05 Finite simple groups and their classification
20D06 Simple groups: alternating groups and groups of Lie type
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[1] J. H. Abawajy, A. V. Kelarev and M. Chowdhury, Power graphs: a survey,Electron. J. Graph Theory Appl. (EJGTA), 1(2) (2013), 125-147. · Zbl 1306.05090
[2] N. Akbari and A. R. Ashrafi, Note on the power graph of finite simple groups, Quasigroups Related Systems, 23(2) (2015), 165-173. · Zbl 1345.20018
[3] A. Ballester-Bolinches, J. Cossey and R. Esteban-Romero, A characterization via graphs of the soluble groups in which permutability is transitive,Algebra Discrete Math., 8(4)(2009), 10-17. · Zbl 1199.20028
[4] S. Bera, On the intersection power graph of a finite group,Electron. J. Graph Theory Appl. (EJGTA), 6(1) (2018), 178-189. · Zbl 1467.05106
[5] S. Bera and A. K. Bhuniya, On enhanced power graphs of finite groups,J. Algebra Appl., 17 (8)(2018), 1850146, 8 pp. · Zbl 1392.05053
[6] A. K. Bhuniya and S. Bera, Normal subgroup based power graphs of a finite group, Comm. Algebra, 45(8) (2017), 3251-3259. · Zbl 1368.05066
[7] A. K. Bhuniya and S. Bera, On some characterizations of strong power graphs of finite groups,Spec. Matrices, 4 (2016), 121-129. · Zbl 1331.05136
[8] N. Biggs,Algebraic Graph Theory, Cambridge University Press, London, 1974. · Zbl 0284.05101
[9] A. Brandst¨adt, Partitions of graphs into one or two independent sets and cliques, Discrete Math., 152(1-3) (1996), 47-54. · Zbl 0853.68140
[10] J. R. Britnell and N. Gill, Perfect commuting graphs,J. Group Theory, 20(1) (2017), 71-—102. · Zbl 1376.20029
[11] D. Bubboloni, M. A. Iranmanesh, S. M. Shaker. Quotient graphs for power graphs, Rend. Semin. Mat. Univ. Padova, 138 (2017), 61-89. · Zbl 1387.05109
[12] P. J. Cameron, The power graph of a finite group. II,J. Group Theory, 13(6) (2010), 779-783. · Zbl 1206.20023
[13] P. J. Cameron, H. Guerra and S. Jurina, The power graph of a torsion-free group, J. Algebraic Combin., 49(1) (2019), 83-98. · Zbl 1410.05085
[14] I. Chakrabarty, S. Ghosh and M. K. Sen, Undirected power graphs of semigroups, Semigroup Forum, 78 (2009), 410-426. · Zbl 1207.05075
[15] S. Chattopadhyay and P. Panigrahi, Some structural properties of power graphs andk-power graphs of finite semigroups,J. Discrete Math. Sci. Cryptogr., 20(5) · Zbl 1495.05125
[16] S. Chattopadhyay and P. Panigrahi, Some relations between power graphs and Cayley graphs,J. Egyptian Math. Soc., 23(3) (2015), 457-462. · Zbl 1328.05079
[17] S. Chattopadhyay and P. Panigrahi, Connectivity and planarity of power graphs of finite cyclic, dihedral and dicyclic groups,Algebra Discrete Math., 18(1) (2014), 42-49. · Zbl 1319.05066
[18] S. Chattopadhyay, P. Panigrahi and F. Atik, Spectral radius of power graphs on certain finite groups,Indag. Math. (N.S.), 29(2) (2018), 730-737. · Zbl 1382.05042
[19] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson R A, Atlas of Finite Groups, Oxford Clarendon Press, 1985. · Zbl 0568.20001
[20] B. Curtin, G. R. Pourgholi and H. Yousefi-Azari, On the punctured power graph of a finite group,Australas. J. Combin., 62 (2015), 1-7. · Zbl 1321.05107
[21] A. K. Das and D. Nongsiang, On the genus of the commuting graphs of finite nonabelian groups,Int. Electron. J. Algebra, 19 (2016), 91-109. · Zbl 1339.20022
[22] A. Doostabadi and M. Farrokhi D. G., Embeddings of (proper) power graphs of finite groups,Algebra Discrete Math., 24(2) (2017), 221-234. · Zbl 1388.05083
[23] A. Doostabadi, M. Farrokhi D. Ghouchan, On the connectivity of proper power graphs of finite groups,Comm. Algebra, 43(10) (2015), 4305-4319. · Zbl 1323.05065
[24] S. F¨oldes and P. L. Hammer, Split graphs,Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La.,1977), 311-315. · Zbl 0407.05071
[25] A. Hamzeh, Signless and normalized Laplacian spectrums of the power graph and its supergraphs of certain finite groups,J. Indones. Math. Soc., 24(1) (2018), 61-69. · Zbl 1451.05105
[26] A. Hamzeh and A. R. Ashrafi,The order supergraph of the power graph of a finite group,Turkish J. Math., 42(4) (2018), 1978-1989. · Zbl 1424.05135
[27] A. Hamzeh and A. R. Ashrafi, Spectrum and L-spectrum of the power graph and its main supergraph for certain finite groups,Filomat, 31(16) (2017), 5323-5334. · Zbl 1499.05362
[28] A. Hamzeh and A. R. Ashrafi, Automorphism groups of supergraphs of the power graph of a finite group,European J. Combin., 60 (2017), 82-88. · Zbl 1348.05095
[29] B. Huppert and N. Blackbrun,Finite Groups II, Springer-Verlag, Berlin, 1982. · Zbl 0477.20001
[30] B. Huppert and N. Blackbrun,Finite Groups III, Springer-Verlag, Berlin, 1982. · Zbl 0514.20002
[31] S. H. Jafari, Some results in a new power graphs in finite groups,Util. Math., 103 (2017), 181-187. · Zbl 1370.20026
[32] S. H. Jafari, Some properties of power graphs in finite group,Asian-Eur. J. Math., 9 (4)(2016), 1650079, 6 pp. · Zbl 1367.20022
[33] S. Kirkland, A. R. Moghaddamfar, S. Navid Salehy, S. Nima Salehy and M. Zohourattar, The complexity of power graphs associated with finite groups, Contributions to Discrete Mathematics, 13(2)(2018), 124-136. · Zbl 1504.05128
[34] A. V. Kelarev, Graph Algebras and Automata, Marcel Dekker, New York, 2003. · Zbl 1070.68097
[35] A. V. Kelarev, Ring Constructions and Applications, World Scientific, River Edge, NJ, 2002. · Zbl 0999.16036
[36] A. V. Kelarev and S. J. Quinn, A combinatorial property and power graphs of groups,The Vienna Conference, Contrib. General Algebra, 12 (2000), 229-235. · Zbl 0966.05040
[37] A. V. Kelarev and S. J. Quinn, Directed graphs and combinatorial properties of semigroups,J. Algebra, 251 (2002), 16-26. · Zbl 1005.20043
[38] A. V. Kelarev and S. J. Quinn, A combinatorial property and power graphs of semigroups,Comment. Math. Univ. Carolinae, 45 (2004), 1-7. · Zbl 1099.05042
[39] A. V. Kelarev, S. J. Quinn and R. Smolikova, Power graphs and semigroups of matrices,Bull. Austral. Math. Soc., 63 (2001), 341-344. · Zbl 1043.20042
[40] A. Kelarev, J. Ryan, J. Yearwood, Cayley graphs as classifiers for data mining: The influence of asymmetries,Discrete Mathematics, 309 (2009), 5360-5369. · Zbl 1206.05050
[41] X. Ma and M. Feng, On the chromatic number of the power graph of a finite group,Indag. Math. (N.S.), 26(4) (2015), 626-633. · Zbl 1317.05063
[42] X. Ma, M. Feng and K. Wang, The rainbow connection number of the power graph of a finite group,Graphs Combin., 32(4) (2016), 1495-1504. · Zbl 1342.05063
[43] X. Ma, R. Fu and X. Lu, On the independence number of the power graph of a finite group,Indag. Math. (N.S.), 29(2) (2018), 794-806. · Zbl 1382.05053
[44] X. Ma, R. Fu, X. Lu, M. Guoand Z. Zhao, Perfect codes in power graphs of finite groups,Open Math., 15 (2017), 1440-1449. · Zbl 1386.05088
[45] A. Mahmoudifar and A. R. Moghaddamfar, Commuting graphs of groups and related numerical parameters,Comm. Algebra, 45(7)(2017), 3159-3165. · Zbl 1368.05069
[46] S. K. Maity, Bipartite and planar power graphs of finite groups,Southeast Asian Bull. Math., 39(4) (2015), 539-543. · Zbl 1340.05124
[47] Z. Mehranian, A. Gholami and A. R. Ashrafi, The spectra of power graphs of certain finite groups,Linear Multilinear Algebra, 65(5) (2017), 1003-1010. · Zbl 1360.05078
[48] Z. Mehranian, A. Gholami and A. R. Ashrafi, A note on the power graph of a finite group,Int. J. Group Theory, 5(1) (2016), 1-10. · Zbl 1454.20049
[49] R. Merris, Laplacian graph eigenvectors,Linear Algebra Appl., 278 (1998), 221-236. · Zbl 0932.05057
[50] A. R. Moghaddamfar, S. Rahbariyan, S. Navid Salehy and S. Nima Salehy, The number of spanning trees of power graphs associated with specific groups and some applications,Ars Combinatoria, 113 (2017), 269-296. · Zbl 1488.05058
[51] A. R. Moghaddamfar, S. Rahbariyan, and W. J. Shi, Certain properties of the power graph associated with a finite group,J. Algebra Appl., 13(7) (2014), 1450040, 18 pp. · Zbl 1304.20025
[52] R. P. Panda and K. V. Krishna, On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups,Comm. Algebra, 46(7) (2018), 3182-3197. · Zbl 1390.05117
[53] R. P. Panda and K. V. Krishna, On connectedness of power graphs of finite groups, J. Algebra Appl., 17(10) (2018), 1850184, 20 pp. · Zbl 1401.05140
[54] K. Pourghobadi and S. H. Jafari, The diameter of power graphs of symmetric groups,J. Algebra Appl., 17(12) (2018), 1850234, 11 pp. · Zbl 1481.20007
[55] G. R. Pourgholi, H. Yousefi-Azari and A. R. Ashrafi, The undirected power graph of a finite group,Bull. Malays. Math. Sci. Soc., 38(4) (2015), 1517-1525. · Zbl 1326.20025
[56] I. V. Protasov and K. D. Protasova, Automorphisms of kaleidoscopical graphs, Algebra Discrete Math., 6(2)(2007), 125-129. · Zbl 1164.05358
[57] H. Sachs, On the number of spanning trees,Proceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975), pp. 529-535. Congressus · Zbl 0324.05102
[58] M. Shaker and M. A. Iranmanesh, On groups with specified quotient power graphs, Int. J. Group Theory, 5(3) (2016), 49-60. · Zbl 1454.20052
[59] Y. Shitov, Coloring the power graph of a semigroup,Graphs Combin., 33(2) (2017), 485-487. · Zbl 1368.05056
[60] A. J. Slupik and V. I. Sushchansky, Minimal generating sets and Cayley graphs of Sylowp-subgroups of finite symmetric groups,Algebra Discrete Math., 8(4)(2009), 167-184. · Zbl 1199.20003
[61] M. Suzuki, A new type of simple groups of finite order,Proc. Nat. Acad. Sci. U.S.A., 46 (1960), 868-870. · Zbl 0093.02301
[62] M. Suzuki, On a class of doubly transitive groups,Ann. of Math., 75 (1) (1962), 105-145. · Zbl 0106.24702
[63] T. Tamizh Chelvam and M. Sattanathan, Power graph of finite abelian groups, Algebra Discrete Math., 16(1) (2013), 33-41. · Zbl 1360.05073
[64] A. V. Zavarnitsine, Finite simple groups with narrow prime spectrum,Sib. Elektron. Mat. Izv., 6 (2009) · Zbl 1289.20021
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