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 Sib. Èlektron. Mat. Izv., 2016, Volume 13, Pages 89–100 (Mi semr658)

Mathematical logic, algebra and number theory

On the realizability of a graph as the Gruenberg–Kegel graph of a finite group

N. V. Maslovaab, D. Pagonc

a N. N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Science, 16, S. Kovalevskaja St, 620990, Ekaterinburg, Russia
b Ural Federal University named after the first President of Russia B. N. Yeltsin, 19, Mira St, 620002, Ekaterinburg, Russia
c University of Maribor, 160, Koroška cesta, 2000, Maribor, Slovenia

Abstract: Let $G$ be a finite group. Denote by $\pi(G)$ the set of all prime divisors of the order of $G$ and by $\omega (G)$ the spectrum of $G$, i.e. the set of all its element orders. The set $\omega(G)$ defines the Gruenberg–Kegel graph (or the prime graph) $\Gamma(G)$ of $G$; in this graph the vertex set is $\pi(G)$ and different vertices $p$ and $q$ are adjacent if and only if $pq\in\omega (G)$. We say that a graph $\Gamma$ with $|\pi(G)|$ vertices is realizable as the Gruenberg–Kegel graph of a group $G$ if there exists a vertices marking of $\Gamma$ by distinct primes from $\pi(G)$ such that the marked graph is equal to $\Gamma(G)$. A graph $\Gamma$ is realizable as the Gruenberg–Kegel graph of a group if $\Gamma$ is realizable as the Gruenberg–Kegel graph of an appropriate group $G$. We prove that a complete bipartite graph $K_{m,n}$ is realizable as the Gruenberg–Kegel graph of a group if and only if $m+n \le 6$ and $(m,n)\not =(3,3)$. Moreover, we describe all the groups $G$ such that the graph $K_{1,5}$ is realizable as the Gruenberg–Kegel graph of $G$.

Keywords: finite group, Gruenberg–Kegel graph (prime graph), realizability of a graph, complete bipartite graph.

 Funding Agency Grant Number Russian Science Foundation 14-11-00061 Dynasty Foundation The work is supported by Russian Science Foundation (project 14-11-00061). The first author is a winner of the competition of the Dmitry Zimin Foundation «Dynasty» for support of young mathematicians in 2013 year.

DOI: https://doi.org/10.17377/semi.2016.13.007

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Bibliographic databases:

Document Type: Article
UDC: 512.54
MSC: 20D60, 05C25, 20C20
Received December 1, 2015, published February 22, 2016
Language: English

Citation: N. V. Maslova, D. Pagon, “On the realizability of a graph as the Gruenberg–Kegel graph of a finite group”, Sib. Èlektron. Mat. Izv., 13 (2016), 89–100

Citation in format AMSBIB
\Bibitem{MasPag16} \by N.~V.~Maslova, D.~Pagon \paper On the realizability of a graph as the Gruenberg--Kegel graph of a finite group \jour Sib. \Elektron. Mat. Izv. \yr 2016 \vol 13 \pages 89--100 \mathnet{http://mi.mathnet.ru/semr658} \crossref{https://doi.org/10.17377/semi.2016.13.007} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000407781100008} `