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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Main Authors: Mohammed Aljohani, Salama Nagy Daoud
Format: Article
Language:English
Published: MDPI AG 2024-12-01
Series:Axioms
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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.
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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
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