On the intersection power graph of a finite group
<p>Given a group <span class="math"><em>G</em></span>, the intersection power graph of <span class="math"><em>G</em></span>, denoted by <span class="math">G<sub><em>I</em></sub>(<em&g...
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| Format: | Article |
| Language: | English |
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Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia
2018-04-01
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| Series: | Electronic Journal of Graph Theory and Applications |
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| Online Access: | https://www.ejgta.org/index.php/ejgta/article/view/465 |
| _version_ | 1818216177465294848 |
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| author | Sudip Bera |
| author_facet | Sudip Bera |
| author_sort | Sudip Bera |
| collection | DOAJ |
| description | <p>Given a group <span class="math"><em>G</em></span>, the intersection power graph of <span class="math"><em>G</em></span>, denoted by <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span>, is the graph with vertex set <span class="math"><em>G</em></span> and two distinct vertices <span class="math"><em>x</em></span> and <span class="math"><em>y</em></span> are adjacent in <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span> if there exists a non-identity element <span class="math"><em>z</em> ∈ <em>G</em></span> such that x<sup>m</sup>=z=y<sup>n</sup>, for some <span class="math"><em>m</em>, <em>n</em> ∈ N</span>, i.e. <span class="math"><em>x</em> ∼ <em>y</em></span> in <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span> if <span class="math">⟨<em>x</em>⟩ ∩ ⟨<em>y</em>⟩ ≠ {<em>e</em>}</span> and <span class="math"><em>e</em></span> is adjacent to all other vertices, where <span class="math"><em>e</em></span> is the identity element of the group <span class="math"><em>G</em></span>. Here we show that the graph <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span> is complete if and only if either <span class="math"><em>G</em></span> is cyclic <span class="math"><em>p</em></span>-group or <span class="math"><em>G</em></span> is a generalized quaternion group. Furthermore, <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span> is Eulerian if and only if <span class="math">∣<em>G</em>∣</span> is odd. We characterize all abelian groups and also all non-abelian <span class="math"><em>p</em></span>-groups <span class="math"><em>G</em></span>, for which <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span> is dominatable. Beside, we determine the automorphism group of the graph <span class="math">G<sub><em>I</em></sub>(Z<sub><em>n</em></sub>)</span>, when <span class="math"><em>n</em> ≠ <em>p</em><sup><em>m</em></sup></span>.</p> |
| first_indexed | 2024-12-12T06:47:50Z |
| format | Article |
| id | doaj.art-bf1ae0dfd70946cb94aaf15449cca53b |
| institution | Directory Open Access Journal |
| issn | 2338-2287 |
| language | English |
| last_indexed | 2024-12-12T06:47:50Z |
| publishDate | 2018-04-01 |
| publisher | Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia |
| record_format | Article |
| series | Electronic Journal of Graph Theory and Applications |
| spelling | doaj.art-bf1ae0dfd70946cb94aaf15449cca53b2022-12-22T00:34:08ZengIndonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), IndonesiaElectronic Journal of Graph Theory and Applications2338-22872018-04-016110.5614/ejgta.2018.6.1.13114On the intersection power graph of a finite groupSudip Bera0Department of Mathematics, Visva-Bharati, Santiniketan-731235, India.<p>Given a group <span class="math"><em>G</em></span>, the intersection power graph of <span class="math"><em>G</em></span>, denoted by <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span>, is the graph with vertex set <span class="math"><em>G</em></span> and two distinct vertices <span class="math"><em>x</em></span> and <span class="math"><em>y</em></span> are adjacent in <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span> if there exists a non-identity element <span class="math"><em>z</em> ∈ <em>G</em></span> such that x<sup>m</sup>=z=y<sup>n</sup>, for some <span class="math"><em>m</em>, <em>n</em> ∈ N</span>, i.e. <span class="math"><em>x</em> ∼ <em>y</em></span> in <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span> if <span class="math">⟨<em>x</em>⟩ ∩ ⟨<em>y</em>⟩ ≠ {<em>e</em>}</span> and <span class="math"><em>e</em></span> is adjacent to all other vertices, where <span class="math"><em>e</em></span> is the identity element of the group <span class="math"><em>G</em></span>. Here we show that the graph <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span> is complete if and only if either <span class="math"><em>G</em></span> is cyclic <span class="math"><em>p</em></span>-group or <span class="math"><em>G</em></span> is a generalized quaternion group. Furthermore, <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span> is Eulerian if and only if <span class="math">∣<em>G</em>∣</span> is odd. We characterize all abelian groups and also all non-abelian <span class="math"><em>p</em></span>-groups <span class="math"><em>G</em></span>, for which <span class="math">G<sub><em>I</em></sub>(<em>G</em>)</span> is dominatable. Beside, we determine the automorphism group of the graph <span class="math">G<sub><em>I</em></sub>(Z<sub><em>n</em></sub>)</span>, when <span class="math"><em>n</em> ≠ <em>p</em><sup><em>m</em></sup></span>.</p>https://www.ejgta.org/index.php/ejgta/article/view/465automorphism group, intersection power graph, planar, p-groups |
| spellingShingle | Sudip Bera On the intersection power graph of a finite group Electronic Journal of Graph Theory and Applications automorphism group, intersection power graph, planar, p-groups |
| title | On the intersection power graph of a finite group |
| title_full | On the intersection power graph of a finite group |
| title_fullStr | On the intersection power graph of a finite group |
| title_full_unstemmed | On the intersection power graph of a finite group |
| title_short | On the intersection power graph of a finite group |
| title_sort | on the intersection power graph of a finite group |
| topic | automorphism group, intersection power graph, planar, p-groups |
| url | https://www.ejgta.org/index.php/ejgta/article/view/465 |
| work_keys_str_mv | AT sudipbera ontheintersectionpowergraphofafinitegroup |