Characterizing finite groups whose enhanced power graphs have universal vertices

David G. Costanzo; Mark L. Lewis; Stefano Schmidt; Eyob Tsegaye; Gabe Udell

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 2, page 637-645
  • ISSN: 0011-4642

Abstract

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Let G be a finite group and construct a graph Δ ( G ) by taking G ∖ { 1 } as the vertex set of Δ ( G ) and by drawing an edge between two vertices x and y if 〈 x , y 〉 is cyclic. Let K ( G ) be the set consisting of the universal vertices of Δ ( 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 Δ ( G ) and K ( G ) when | G | is divisible by two distinct primes and the diameter of Δ ( G ) is 2.

How to cite

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Costanzo, David G., et al. "Characterizing finite groups whose enhanced power graphs have universal vertices." Czechoslovak Mathematical Journal 74.2 (2024): 637-645. <http://eudml.org/doc/299372>.

@article{Costanzo2024,
abstract = {Let $G$ be a finite group and construct a graph $\Delta (G)$ by taking $G\setminus \lbrace 1\rbrace $ 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.},
author = {Costanzo, David G., Lewis, Mark L., Schmidt, Stefano, Tsegaye, Eyob, Udell, Gabe},
journal = {Czechoslovak Mathematical Journal},
keywords = {enhanced power graph; universal vertex; diameter},
language = {eng},
number = {2},
pages = {637-645},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Characterizing finite groups whose enhanced power graphs have universal vertices},
url = {http://eudml.org/doc/299372},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Costanzo, David G.
AU - Lewis, Mark L.
AU - Schmidt, Stefano
AU - Tsegaye, Eyob
AU - Udell, Gabe
TI - Characterizing finite groups whose enhanced power graphs have universal vertices
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 2
SP - 637
EP - 645
AB - Let $G$ be a finite group and construct a graph $\Delta (G)$ by taking $G\setminus \lbrace 1\rbrace $ 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.
LA - eng
KW - enhanced power graph; universal vertex; diameter
UR - http://eudml.org/doc/299372
ER -

References

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