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Characterization of commuting graphs of finite groups having small genus

Characterization of commuting graphs of finite groups having small genus

In this paper we first show that among all double-toroidal and triple-toroidal finite graphs only \(K_8 \sqcup 9K_1\), \(K_8 \sqcup 5K_2\), \(K_8 \sqcup 3K_4\), \(K_8 \sqcup 9K_3\), \(K_8\sqcup 9(K_1 \vee 3K_2)\), \(3K_6\) and \(3K_6 \sqcup 4K_4 \sqcup 6K_2\) can be realized as commuting graphs of f...

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Main Authors: Das, Shrabani, Nongsiang, Deiborlang, Nath, Rajat Kanti
Format: Article
Language:English
Published: Lugansk National Taras Shevchenko University 2024
Subjects:
commuting graph
genus
planar graph
double-toroidal
triple-toroidal
20D60
05C25
05C09
Online Access:https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2211
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spelling oai:ojs.admjournal.luguniv.edu.ua:article-22112024-06-27T08:42:43Z Characterization of commuting graphs of finite groups having small genus Das, Shrabani Nongsiang, Deiborlang Nath, Rajat Kanti commuting graph, genus, planar graph, double-toroidal, triple-toroidal 20D60, 05C25, 05C09 In this paper we first show that among all double-toroidal and triple-toroidal finite graphs only \(K_8 \sqcup 9K_1\), \(K_8 \sqcup 5K_2\), \(K_8 \sqcup 3K_4\), \(K_8 \sqcup 9K_3\), \(K_8\sqcup 9(K_1 \vee 3K_2)\), \(3K_6\) and \(3K_6 \sqcup 4K_4 \sqcup 6K_2\) can be realized as commuting graphs of finite groups, where \(\sqcup\) and \(\vee\) stand for disjoint union and join of graphs respectively. As consequences of our results we also show that for any finite non-abelian group \(G\) if the commuting graph of \(G\) (denoted by \(\Gamma_c(G)\)) is double-toroidal or  triple-toroidal then \(\Gamma_c(G)\) and its complement satisfy Hansen-Vukičević Conjecture and E-LE conjecture. In the process we find a non-complete graph, namely the non-commuting graph of the group \((\mathbb{Z}_3 \times \mathbb{Z}_3) \rtimes Q_8\), that is hyperenergetic. This gives a new counter example to a conjecture of Gutman regarding hyperenergetic graphs. Lugansk National Taras Shevchenko University 2024-06-27 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2211 10.12958/adm2211 Algebra and Discrete Mathematics; Vol 37, No 2 (2024) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2211/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2211/1183 Copyright (c) 2024 Algebra and Discrete Mathematics
institution Algebra and Discrete Mathematics
baseUrl_str
datestamp_date 2024-06-27T08:42:43Z
collection OJS
language English
topic commuting graph
genus
planar graph
double-toroidal
triple-toroidal
20D60
05C25
05C09
spellingShingle commuting graph
genus
planar graph
double-toroidal
triple-toroidal
20D60
05C25
05C09
Das, Shrabani
Nongsiang, Deiborlang
Nath, Rajat Kanti
Characterization of commuting graphs of finite groups having small genus
topic_facet commuting graph
genus
planar graph
double-toroidal
triple-toroidal
20D60
05C25
05C09
format Article
author Das, Shrabani
Nongsiang, Deiborlang
Nath, Rajat Kanti
author_facet Das, Shrabani
Nongsiang, Deiborlang
Nath, Rajat Kanti
author_sort Das, Shrabani
title Characterization of commuting graphs of finite groups having small genus
title_short Characterization of commuting graphs of finite groups having small genus
title_full Characterization of commuting graphs of finite groups having small genus
title_fullStr Characterization of commuting graphs of finite groups having small genus
title_full_unstemmed Characterization of commuting graphs of finite groups having small genus
title_sort characterization of commuting graphs of finite groups having small genus
description In this paper we first show that among all double-toroidal and triple-toroidal finite graphs only \(K_8 \sqcup 9K_1\), \(K_8 \sqcup 5K_2\), \(K_8 \sqcup 3K_4\), \(K_8 \sqcup 9K_3\), \(K_8\sqcup 9(K_1 \vee 3K_2)\), \(3K_6\) and \(3K_6 \sqcup 4K_4 \sqcup 6K_2\) can be realized as commuting graphs of finite groups, where \(\sqcup\) and \(\vee\) stand for disjoint union and join of graphs respectively. As consequences of our results we also show that for any finite non-abelian group \(G\) if the commuting graph of \(G\) (denoted by \(\Gamma_c(G)\)) is double-toroidal or  triple-toroidal then \(\Gamma_c(G)\) and its complement satisfy Hansen-Vukičević Conjecture and E-LE conjecture. In the process we find a non-complete graph, namely the non-commuting graph of the group \((\mathbb{Z}_3 \times \mathbb{Z}_3) \rtimes Q_8\), that is hyperenergetic. This gives a new counter example to a conjecture of Gutman regarding hyperenergetic graphs.
publisher Lugansk National Taras Shevchenko University
publishDate 2024
url https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2211
work_keys_str_mv AT dasshrabani characterizationofcommutinggraphsoffinitegroupshavingsmallgenus
AT nongsiangdeiborlang characterizationofcommutinggraphsoffinitegroupshavingsmallgenus
AT nathrajatkanti characterizationofcommutinggraphsoffinitegroupshavingsmallgenus
first_indexed 2024-06-28T04:03:55Z
last_indexed 2024-06-28T04:03:55Z
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