Character graphs with diameter three

Ebrahimi, Mahdi

doi:10.1090/proc/15160

Character graphs with diameter three
HTML articles powered by AMS MathViewer

by Mahdi Ebrahimi PDF

Proc. Amer. Math. Soc. 148 (2020), 4615-4619 Request permission

Abstract:

For a finite group $G$, let $\Delta (G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. In this paper, we show that if the diameter of $\Delta (G)$ is equal to three, then the complement of $\Delta (G)$ is bipartite. Also in this case, we determine the structure of the character graph $\Delta (G)$.

References

Zeinab Akhlaghi, Carlo Casolo, Silvio Dolfi, Khatoon Khedri, and Emanuele Pacifici, On the character degree graph of solvable groups, Proc. Amer. Math. Soc. 146 (2018), no. 4, 1505–1513. MR 3754337, DOI 10.1090/proc/13879
Zeinab Akhlaghi, Carlo Casolo, Silvio Dolfi, Emanuele Pacifici, and Lucia Sanus, On the character degree graph of finite groups, Ann. Mat. Pura Appl. (4) 198 (2019), no. 5, 1595–1614. MR 4022111, DOI 10.1007/s10231-019-00833-0
Yakov Berkovich, Finite groups with small sums of degrees of some non-linear irreducible characters, J. Algebra 171 (1995), no. 2, 426–443. MR 1315905, DOI 10.1006/jabr.1995.1020
Mahdi Ebrahimi, $K_4$-free character graphs with seven vertices, Comm. Algebra 48 (2020), no. 3, 1001–1010. MR 4079511, DOI 10.1080/00927872.2019.1670197
Mahdi Ebrahimi, Ali Iranmanesh, and Mohammad Ali Hosseinzadeh, Hamiltonian character graphs, J. Algebra 428 (2015), 54–66. MR 3314285, DOI 10.1016/j.jalgebra.2014.12.038
Bertram Huppert, Some simple groups which are determined by the set of their character degrees. I, Illinois J. Math. 44 (2000), no. 4, 828–842. MR 1804317
I. Martin Isaacs, Character theory of finite groups, AMS Chelsea Publishing, Providence, RI, 2006. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423]. MR 2270898, DOI 10.1090/chel/359
Mark L. Lewis, Solvable groups with character degree graphs having 5 vertices and diameter 3, Comm. Algebra 30 (2002), no. 11, 5485–5503. MR 1945100, DOI 10.1081/AGB-120015664
Mark L. Lewis, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, Rocky Mountain J. Math. 38 (2008), no. 1, 175–211. MR 2397031, DOI 10.1216/RMJ-2008-38-1-175
Mark L. Lewis and Donald L. White, Diameters of degree graphs of nonsolvable groups. II, J. Algebra 312 (2007), no. 2, 634–649. MR 2333176, DOI 10.1016/j.jalgebra.2007.02.057
Mark L. Lewis and Donald L. White, Nonsolvable groups with no prime dividing three character degrees, J. Algebra 336 (2011), 158–183. MR 2802535, DOI 10.1016/j.jalgebra.2011.03.028
O. Manz, R. Staszewski, and W. Willems, On the number of components of a graph related to character degrees, Proc. Amer. Math. Soc. 103 (1988), no. 1, 31–37. MR 938639, DOI 10.1090/S0002-9939-1988-0938639-1
Olaf Manz, Wolfgang Willems, and Thomas R. Wolf, The diameter of the character degree graph, J. Reine Angew. Math. 402 (1989), 181–198. MR 1022799
Olaf Manz and Thomas R. Wolf, Representations of solvable groups, London Mathematical Society Lecture Note Series, vol. 185, Cambridge University Press, Cambridge, 1993. MR 1261638, DOI 10.1017/CBO9780511525971
Z. Sayanjali, Z. Akhlaghi, and B. Khosravi, On the regularity of character degree graphs, Bull. Aust. Math. Soc. 100 (2019), no. 3, 428–433. MR 4028190, DOI 10.1017/s0004972719000315
Hung P. Tong-Viet, Groups whose prime graphs have no triangles, J. Algebra 378 (2013), 196–206. MR 3017021, DOI 10.1016/j.jalgebra.2012.12.024
Donald L. White, Degree graphs of simple linear and unitary groups, Comm. Algebra 34 (2006), no. 8, 2907–2921. MR 2250577, DOI 10.1080/00927870600639419