Czechoslovak Mathematical Journal, first online, pp. 1-9
Characterizing finite groups whose enhanced power graphs have universal vertices
David G. Costanzo, Mark L. Lewis, Stefano Schmidt, Eyob Tsegaye, Gabe Udell
Received February 14, 2024. Published online April 29, 2024.
Abstract: 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.
Keywords: enhanced power graph; universal vertex; diameter
Affiliations: David G. Costanzo, School of Mathematical and Statistical Sciences, O-110 Martin Hall, Box 340975, Clemson University, Clemson, SC 29634, USA e-mail: davidgcostanzo@gmail.com; Mark L. Lewis (corresponding author), Department of Mathematical Sciences, 1300 Lefton Esplanade, Kent State University, Kent, OH 44242, USA, e-mail: lewis@math.kent.edu; Stefano Schmidt, Eyob Tsegaye, Department of Mathematics, Princeton University, Fine Hall, 304 Washington Rd, Princeton, NJ 08540, USA, e-mail: eyob@princeton.edu; Gabe Udell, Department of Mathematics, Malott Hall, 212 Garden Ave, Cornell University, Ithaca, NY 14853, USA, e-mail: gru5@cornell.edu
