On combinatorial properties of Gruenberg–Kegel graphs of finite groups
If G is a finite group, then the spectrum ω ( G ) is the set of all element orders of G . The prime spectrum π ( G ) is the set of all primes belonging to ω ( G ) . A simple graph Γ ( G ) whose vertex set is π ( G ) and in which two distinct vertices r and s are adjacent if and only if r s ∈ ω ( G )...
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| Veröffentlicht in: | Monatshefte für Mathematik 2024-12, Vol.205 (4), p.711-723 |
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| container_title | Monatshefte für Mathematik |
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| creator | Chen, Mingzhu Gorshkov, Ilya Maslova, Natalia V. Yang, Nanying |
| description | If
G
is a finite group, then the spectrum
ω
(
G
)
is the set of all element orders of
G
. The prime spectrum
π
(
G
)
is the set of all primes belonging to
ω
(
G
)
. A simple graph
Γ
(
G
)
whose vertex set is
π
(
G
)
and in which two distinct vertices
r
and
s
are adjacent if and only if
r
s
∈
ω
(
G
)
is called the Gruenberg–Kegel graph or the prime graph of
G
. In this paper, we prove that if
G
is a group of even order, then the set of vertices which are non-adjacent to 2 in
Γ
(
G
)
forms a union of cliques. Moreover, we decide when a strongly regular graph is isomorphic to the Gruenberg–Kegel graph of a finite group. |
| doi_str_mv | 10.1007/s00605-024-02005-6 |
| format | Article |
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G
is a finite group, then the spectrum
ω
(
G
)
is the set of all element orders of
G
. The prime spectrum
π
(
G
)
is the set of all primes belonging to
ω
(
G
)
. A simple graph
Γ
(
G
)
whose vertex set is
π
(
G
)
and in which two distinct vertices
r
and
s
are adjacent if and only if
r
s
∈
ω
(
G
)
is called the Gruenberg–Kegel graph or the prime graph of
G
. In this paper, we prove that if
G
is a group of even order, then the set of vertices which are non-adjacent to 2 in
Γ
(
G
)
forms a union of cliques. Moreover, we decide when a strongly regular graph is isomorphic to the Gruenberg–Kegel graph of a finite group.</description><identifier>ISSN: 0026-9255</identifier><identifier>EISSN: 1436-5081</identifier><identifier>DOI: 10.1007/s00605-024-02005-6</identifier><language>eng</language><publisher>Vienna: Springer Vienna</publisher><subject>Apexes ; Combinatorial analysis ; Graph theory ; Group theory ; Mathematics ; Mathematics and Statistics ; Vertex sets</subject><ispartof>Monatshefte für Mathematik, 2024-12, Vol.205 (4), p.711-723</ispartof><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c200t-af3ed6d2929b37b1d42f3d5947b894ef92dde19ae8ec512da36f60e9ff8d6d563</cites><orcidid>0000-0001-6574-5335</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00605-024-02005-6$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00605-024-02005-6$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Chen, Mingzhu</creatorcontrib><creatorcontrib>Gorshkov, Ilya</creatorcontrib><creatorcontrib>Maslova, Natalia V.</creatorcontrib><creatorcontrib>Yang, Nanying</creatorcontrib><title>On combinatorial properties of Gruenberg–Kegel graphs of finite groups</title><title>Monatshefte für Mathematik</title><addtitle>Monatsh Math</addtitle><description>If
G
is a finite group, then the spectrum
ω
(
G
)
is the set of all element orders of
G
. The prime spectrum
π
(
G
)
is the set of all primes belonging to
ω
(
G
)
. A simple graph
Γ
(
G
)
whose vertex set is
π
(
G
)
and in which two distinct vertices
r
and
s
are adjacent if and only if
r
s
∈
ω
(
G
)
is called the Gruenberg–Kegel graph or the prime graph of
G
. In this paper, we prove that if
G
is a group of even order, then the set of vertices which are non-adjacent to 2 in
Γ
(
G
)
forms a union of cliques. Moreover, we decide when a strongly regular graph is isomorphic to the Gruenberg–Kegel graph of a finite group.</description><subject>Apexes</subject><subject>Combinatorial analysis</subject><subject>Graph theory</subject><subject>Group theory</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Vertex sets</subject><issn>0026-9255</issn><issn>1436-5081</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9UEFOwzAQtBBIlMIHOEXiHFjbsRMfUQUtolIvcLacZB1StXGwkwM3_sAPeQlug8SNw2pXuzOzoyHkmsItBcjvAoAEkQLLYkGc5AmZ0YzLVEBBT8kMgMlUMSHOyUUIWwCgXKoZWW26pHL7su3M4HxrdknvXY9-aDEkziZLP2JXom--P7-escFd0njTvx1vtu3aAePCjX24JGfW7AJe_fY5eX18eFms0vVm-bS4X6dV9DWkxnKsZc0UUyXPS1pnzPJaqCwvC5WhVayukSqDBVaCstpwaSWgsraINCH5nNxMutHn-4hh0Fs3-i6-1JwyECIDTiOKTajKuxA8Wt37dm_8h6agD4npKTEdE9PHxPRBmk-kEMFdg_5P-h_WD5ybb7U</recordid><startdate>20241201</startdate><enddate>20241201</enddate><creator>Chen, Mingzhu</creator><creator>Gorshkov, Ilya</creator><creator>Maslova, Natalia V.</creator><creator>Yang, Nanying</creator><general>Springer Vienna</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0001-6574-5335</orcidid></search><sort><creationdate>20241201</creationdate><title>On combinatorial properties of Gruenberg–Kegel graphs of finite groups</title><author>Chen, Mingzhu ; Gorshkov, Ilya ; Maslova, Natalia V. ; Yang, Nanying</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c200t-af3ed6d2929b37b1d42f3d5947b894ef92dde19ae8ec512da36f60e9ff8d6d563</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Apexes</topic><topic>Combinatorial analysis</topic><topic>Graph theory</topic><topic>Group theory</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Vertex sets</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Chen, Mingzhu</creatorcontrib><creatorcontrib>Gorshkov, Ilya</creatorcontrib><creatorcontrib>Maslova, Natalia V.</creatorcontrib><creatorcontrib>Yang, Nanying</creatorcontrib><collection>CrossRef</collection><jtitle>Monatshefte für Mathematik</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Chen, Mingzhu</au><au>Gorshkov, Ilya</au><au>Maslova, Natalia V.</au><au>Yang, Nanying</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On combinatorial properties of Gruenberg–Kegel graphs of finite groups</atitle><jtitle>Monatshefte für Mathematik</jtitle><stitle>Monatsh Math</stitle><date>2024-12-01</date><risdate>2024</risdate><volume>205</volume><issue>4</issue><spage>711</spage><epage>723</epage><pages>711-723</pages><issn>0026-9255</issn><eissn>1436-5081</eissn><abstract>If
G
is a finite group, then the spectrum
ω
(
G
)
is the set of all element orders of
G
. The prime spectrum
π
(
G
)
is the set of all primes belonging to
ω
(
G
)
. A simple graph
Γ
(
G
)
whose vertex set is
π
(
G
)
and in which two distinct vertices
r
and
s
are adjacent if and only if
r
s
∈
ω
(
G
)
is called the Gruenberg–Kegel graph or the prime graph of
G
. In this paper, we prove that if
G
is a group of even order, then the set of vertices which are non-adjacent to 2 in
Γ
(
G
)
forms a union of cliques. Moreover, we decide when a strongly regular graph is isomorphic to the Gruenberg–Kegel graph of a finite group.</abstract><cop>Vienna</cop><pub>Springer Vienna</pub><doi>10.1007/s00605-024-02005-6</doi><tpages>13</tpages><orcidid>https://orcid.org/0000-0001-6574-5335</orcidid></addata></record> |
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| ispartof | Monatshefte für Mathematik, 2024-12, Vol.205 (4), p.711-723 |
| issn | 0026-9255 1436-5081 |
| language | eng |
| recordid | cdi_proquest_journals_3120554031 |
| source | Springer Nature |
| subjects | Apexes Combinatorial analysis Graph theory Group theory Mathematics Mathematics and Statistics Vertex sets |
| title | On combinatorial properties of Gruenberg–Kegel graphs of finite groups |
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