Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
enhanced power graph; universal vertex; diameter
enhanced power graph; universal vertex; diameter
Summary:
Let $G$ be a finite group and construct a graph $\Delta (G)$ by taking $G\setminus \{1\}$ as the vertex set of $\Delta (G)$ and by drawing an edge between two vertices $x$ and $y$ if $\langle x,y\rangle $ is cyclic. Let $K(G)$ be the set consisting of the universal vertices of $\Delta (G)$ along the identity element. For a solvable group $G$, we present a necessary and sufficient condition for $K(G)$ to be nontrivial. We also develop a connection between $\Delta (G)$ and $K(G)$ when $|G|$ is divisible by two distinct primes and the diameter of $\Delta (G)$ is 2.
References:
[1] Aalipour, G., Akbari, S., Cameron, P. J., Nikandish, R., Shaveisi, F.: On the structure of the power graph and the enhanced power graph of a group. Electron. J. Comb. 24 (2017), Article ID P3.16, 18 pages \99999DOI99999 10.37236/6497 . MR 3691533 | Zbl 1369.05059
[2] Abdollahi, A., Hassanabadi, A. Mohammadi: Noncyclic graph of a group. Commun. Algebra 35 (2007), 2057-2081. DOI 10.1080/00927870701302081 | MR 2331830 | Zbl 1131.20016
[3] Bera, S., Bhuniya, A. K.: On enhanced power graphs of finite groups. J. Algebra Appl. 17 (2018), Article ID 1850146, 8 pages. DOI 10.1142/S0219498818501463 | MR 3825307 | Zbl 1392.05053
[4] Bera, S., Dey, H. K.: On the proper enhanced power graphs of finite nilpotent groups. J. Group Theory 25 (2022), 1109-1131. DOI 10.1515/jgth-2022-0057 | MR 4504122 | Zbl 1510.20019
[5] Bera, S., Dey, H. K., Mukherjee, S. K.: On the connectivity of enhanced power graphs of finite groups. Graphs Comb. 37 (2021), 591-603. DOI 10.1007/s00373-020-02267-5 | MR 4221643 | Zbl 1492.05076
[6] Berkovich, Y.: Groups of Prime Power Order. Volume 1. de Gruyter Expositions in Mathematics 46. Walter De Gruyter, Berlin (2008). DOI 10.1515/9783110208221 | MR 2464640 | Zbl 1168.20001
[7] Cameron, P. J.: Graphs defined on groups. Int. J. Group Theory 11 (2022), 53-107. DOI 10.22108/IJGT.2021.127679.1681 | MR 4346241 | Zbl 1496.05070
[8] Costanzo, D. G., Lewis, M. L., Schmidt, S., Tsegaye, E., Udell, G.: The cyclic graph of a $Z$-group. Bull. Aust. Math. Soc. 104 (2021), 295-301. DOI 10.1017/S0004972720001318 | MR 4308146 | Zbl 07394396
[9] Imperatore, D.: On a graph associated with a group. Ischia Group Theory 2008 World Scientific, Hackensack (2009), 100-115. DOI 10.1142/9789814277808_0008 | MR 2816425 | Zbl 1191.20017
[10] Imperatore, D., Lewis, M. L.: A condition in finite solvable groups related to cyclic subgroups. Bull. Aust. Math. Soc. 83 (2011), 267-272. DOI 10.1017/S0004972710001747 | MR 2784785 | Zbl 1220.20015
[11] Isaacs, I. M.: Finite Group Theory. Graduate Studies in Mathematics 92. AMS, Providence (2008). DOI 10.1090/gsm/092 | MR 2426855 | Zbl 1169.20001
[12] Ma, X., Kelarev, A., Lin, Y., Wang, K.: A survey on enhanced power graphs of finite groups. Electron. J. Graph Theory Appl. 10 (2022), 89-111. DOI 10.5614/ejgta.2022.10.1.6 | MR 4446436 | Zbl 1487.05119
[13] Ma, X., She, Y.: The metric dimension of the enhanced power graph of a finite group. J. Algebra Appl. 19 (2020), Article ID 2050020, 14 pages. DOI 10.1142/S0219498820500206 | MR 4065011 | Zbl 1437.05098
[14] Mahmoudifar, A., Babai, A.: On the structure of finite groups with dominatable enhanced power graph. J. Algebra Appl. 21 (2022), Article ID 2250176, 8 pages. DOI 10.1142/S0219498822501766 | MR 4474716 | Zbl 1496.05075
[15] O'Bryant, K., Patrick, D., Smithline, L., Wepsic, E.: Some facts about cycels and tidy groups. Mathematical Sciences Technical Reports (MSTR) 131 (1992), 8 pages Available at \brokenlink { https://scholar.rose-hulman.edu/math_mstr/131}\kern0pt
[16] Suzuki, M.: Group Theory. Volume 2. Grundlehren der Mathematischen Wissenschaften 248. Springer, New York (1986). MR 0815926 | Zbl 0586.20001
[17] Puttkamer, T. W. von: On the Finiteness of the Classifying Space for Virtually Cyclic Subgroups: Dissertation. Rheinischen Friedrich-Wilhelms-Universität, Bonn (2018), Available at https://core.ac.uk/download/pdf/322960861.pdf\kern0pt MR 3950648
Search
Partner of
