The Edge Odd Graceful Labeling of Water Wheel Graphs
A graph, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</...
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MDPI AG
2024-12-01
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Online Access: | https://www.mdpi.com/2075-1680/14/1/5 |
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author | Mohammed Aljohani Salama Nagy Daoud |
author_facet | Mohammed Aljohani Salama Nagy Daoud |
author_sort | Mohammed Aljohani |
collection | DOAJ |
description | A graph, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is edge odd graceful if it possesses edge odd graceful labeling. This labeling is defined as a bijection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>:</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula>, from which an injective transformation is derived, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>g</mi><mo>*</mo></msup><mo>:</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, from the rule that the image of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> under <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>g</mi><mo>*</mo></msup></semantics></math></inline-formula> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∑</mo><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mi>g</mi><mrow><mo>(</mo><mi>u</mi><mi>v</mi><mo>)</mo></mrow><mspace width="0.277778em"></mspace><mi>mod</mi><mspace width="0.277778em"></mspace><mrow><mo>(</mo><mn>2</mn><mi>m</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The main objective of this manuscript is to introduce new classes of planar graphs, namely water wheel graphs, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><msub><mi>W</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>; triangulated water wheel graphs, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><msub><mi>W</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>; closed water wheel graphs, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>W</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>; and closed triangulated water wheel graphs, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>T</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>. Furthermore, we specify conditions for these graphs to allow for edge odd graceful labelings. |
format | Article |
id | doaj-art-9a3cd2ca6b9142f594c473224bc02070 |
institution | Kabale University |
issn | 2075-1680 |
language | English |
publishDate | 2024-12-01 |
publisher | MDPI AG |
record_format | Article |
series | Axioms |
spelling | doaj-art-9a3cd2ca6b9142f594c473224bc020702025-01-24T13:22:07ZengMDPI AGAxioms2075-16802024-12-01141510.3390/axioms14010005The Edge Odd Graceful Labeling of Water Wheel GraphsMohammed Aljohani0Salama Nagy Daoud1Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah 41411, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Taibah University, Al-Madinah 41411, Saudi ArabiaA graph, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is edge odd graceful if it possesses edge odd graceful labeling. This labeling is defined as a bijection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>:</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>→</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula>, from which an injective transformation is derived, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>g</mi><mo>*</mo></msup><mo>:</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>2</mn><mi>m</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, from the rule that the image of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></semantics></math></inline-formula> under <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>g</mi><mo>*</mo></msup></semantics></math></inline-formula> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∑</mo><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mi>g</mi><mrow><mo>(</mo><mi>u</mi><mi>v</mi><mo>)</mo></mrow><mspace width="0.277778em"></mspace><mi>mod</mi><mspace width="0.277778em"></mspace><mrow><mo>(</mo><mn>2</mn><mi>m</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The main objective of this manuscript is to introduce new classes of planar graphs, namely water wheel graphs, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><msub><mi>W</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>; triangulated water wheel graphs, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><msub><mi>W</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>; closed water wheel graphs, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>W</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>; and closed triangulated water wheel graphs, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>T</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>. Furthermore, we specify conditions for these graphs to allow for edge odd graceful labelings.https://www.mdpi.com/2075-1680/14/1/5graceful labelingedge graceful labelingedge odd graceful labelingwater wheel graph |
spellingShingle | Mohammed Aljohani Salama Nagy Daoud The Edge Odd Graceful Labeling of Water Wheel Graphs Axioms graceful labeling edge graceful labeling edge odd graceful labeling water wheel graph |
title | The Edge Odd Graceful Labeling of Water Wheel Graphs |
title_full | The Edge Odd Graceful Labeling of Water Wheel Graphs |
title_fullStr | The Edge Odd Graceful Labeling of Water Wheel Graphs |
title_full_unstemmed | The Edge Odd Graceful Labeling of Water Wheel Graphs |
title_short | The Edge Odd Graceful Labeling of Water Wheel Graphs |
title_sort | edge odd graceful labeling of water wheel graphs |
topic | graceful labeling edge graceful labeling edge odd graceful labeling water wheel graph |
url | https://www.mdpi.com/2075-1680/14/1/5 |
work_keys_str_mv | AT mohammedaljohani theedgeoddgracefullabelingofwaterwheelgraphs AT salamanagydaoud theedgeoddgracefullabelingofwaterwheelgraphs AT mohammedaljohani edgeoddgracefullabelingofwaterwheelgraphs AT salamanagydaoud edgeoddgracefullabelingofwaterwheelgraphs |