Note on structural properties of graphs
In this paper, we establish sufficient and necessary conditions for the existence of abelian subgroups of maximal order of a finite group GG, by means of its commuting graph. The order of these subgroups attains the bound c=∣[x1]∣+⋯+∣[xm]∣c=| \left[{x}_{1}]| \hspace{-0.25em}+\cdots +\hspace{-0.25em}...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2022-02-01
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| Series: | Open Mathematics |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/math-2021-0137 |
| _version_ | 1797998373964349440 |
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| author | Arreola-Bautista Luis D. Reyna Gerardo Romero-Valencia Jesús Sigarreta José M. |
| author_facet | Arreola-Bautista Luis D. Reyna Gerardo Romero-Valencia Jesús Sigarreta José M. |
| author_sort | Arreola-Bautista Luis D. |
| collection | DOAJ |
| description | In this paper, we establish sufficient and necessary conditions for the existence of abelian subgroups of maximal order of a finite group GG, by means of its commuting graph. The order of these subgroups attains the bound c=∣[x1]∣+⋯+∣[xm]∣c=| \left[{x}_{1}]| \hspace{-0.25em}+\cdots +\hspace{-0.25em}\hspace{0.33em}| \left[{x}_{m}]| , where [xi]\left[{x}_{i}] denotes the conjugacy class of xi{x}_{i} in GG and mm is the smallest integer jj such that ∣[x1]∣+⋯+∣[xj]∣≥∣CG(xj)∣| \left[{x}_{1}]| \hspace{-0.25em}+\cdots +| \left[{x}_{j}]| \ge | {C}_{G}\left({x}_{j})| , where CG(xj){C}_{G}\left({x}_{j}) is the centralizer of xj{x}_{j} in GG. |
| first_indexed | 2024-04-11T10:47:41Z |
| format | Article |
| id | doaj.art-7809dc4a81e84206ab28927a3e737d4b |
| institution | Directory Open Access Journal |
| issn | 2391-5455 |
| language | English |
| last_indexed | 2024-04-11T10:47:41Z |
| publishDate | 2022-02-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj.art-7809dc4a81e84206ab28927a3e737d4b2022-12-22T04:29:00ZengDe GruyterOpen Mathematics2391-54552022-02-01201162310.1515/math-2021-0137Note on structural properties of graphsArreola-Bautista Luis D.0Reyna Gerardo1Romero-Valencia Jesús2Sigarreta José M.3Facultad de Matemáticas, Universidad Autónoma de Guerrero, 54, Carlos Adame, La Garita, 39650 Acapulco, Guerrero, MéxicoFacultad de Matemáticas, Universidad Autónoma de Guerrero, 54, Carlos Adame, La Garita, 39650 Acapulco, Guerrero, MéxicoFacultad de Matemáticas, Universidad Autónoma de Guerrero, Sauce 19, La Cima, 39086 Chilpancingo de los Bravo, Guerrero, MéxicoFacultad de Matemáticas, Universidad Autónoma de Guerrero, 54, Carlos Adame, La Garita, 39650 Acapulco, Guerrero, MéxicoIn this paper, we establish sufficient and necessary conditions for the existence of abelian subgroups of maximal order of a finite group GG, by means of its commuting graph. The order of these subgroups attains the bound c=∣[x1]∣+⋯+∣[xm]∣c=| \left[{x}_{1}]| \hspace{-0.25em}+\cdots +\hspace{-0.25em}\hspace{0.33em}| \left[{x}_{m}]| , where [xi]\left[{x}_{i}] denotes the conjugacy class of xi{x}_{i} in GG and mm is the smallest integer jj such that ∣[x1]∣+⋯+∣[xj]∣≥∣CG(xj)∣| \left[{x}_{1}]| \hspace{-0.25em}+\cdots +| \left[{x}_{j}]| \ge | {C}_{G}\left({x}_{j})| , where CG(xj){C}_{G}\left({x}_{j}) is the centralizer of xj{x}_{j} in GG.https://doi.org/10.1515/math-2021-0137abelian subgroupcommuting graphcliquechromatic numberweakly perfect graph05c75 |
| spellingShingle | Arreola-Bautista Luis D. Reyna Gerardo Romero-Valencia Jesús Sigarreta José M. Note on structural properties of graphs Open Mathematics abelian subgroup commuting graph clique chromatic number weakly perfect graph 05c75 |
| title | Note on structural properties of graphs |
| title_full | Note on structural properties of graphs |
| title_fullStr | Note on structural properties of graphs |
| title_full_unstemmed | Note on structural properties of graphs |
| title_short | Note on structural properties of graphs |
| title_sort | note on structural properties of graphs |
| topic | abelian subgroup commuting graph clique chromatic number weakly perfect graph 05c75 |
| url | https://doi.org/10.1515/math-2021-0137 |
| work_keys_str_mv | AT arreolabautistaluisd noteonstructuralpropertiesofgraphs AT reynagerardo noteonstructuralpropertiesofgraphs AT romerovalenciajesus noteonstructuralpropertiesofgraphs AT sigarretajosem noteonstructuralpropertiesofgraphs |