Geodesic
Characterizing finite groups whose enhanced power graphs have universal vertices
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 2, pp. 637-645
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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.
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.
DOI : 10.21136/CMJ.2024.0065-24
Classification : 05C25, 20D25
Keywords: enhanced power graph; universal vertex; diameter 
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     title = {Characterizing finite groups whose enhanced power graphs have universal vertices},
     journal = {Czechoslovak Mathematical Journal},
     pages = {637--645},
     year = {2024},
     volume = {74},
     number = {2},
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     language = {en},
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Costanzo, David G.; Lewis, Mark L.; Schmidt, Stefano; Tsegaye, Eyob; Udell, Gabe. Characterizing finite groups whose enhanced power graphs have universal vertices. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 2, pp. 637-645. doi: 10.21136/CMJ.2024.0065-24

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