
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. Maslova^{ab}, D. Pagon^{c} ^{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 
141100061 
Dynasty Foundation 

The work is supported by Russian Science Foundation (project 141100061). 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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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
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\by N.~V.~Maslova, D.~Pagon
\paper On the realizability of a graph as the GruenbergKegel graph of a finite group
\jour Sib. \`Elektron. Mat. Izv.
\yr 2016
\vol 13
\pages 89100
\mathnet{http://mi.mathnet.ru/semr658}
\crossref{https://doi.org/10.17377/semi.2016.13.007}
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