Modular Version of Edge Irregularity Strength for Fan and Wheel Graphs
A <i>k</i>-labeling from the vertex set of a simple 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>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-12-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/12/2671 |
| _version_ | 1797455161806815232 |
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| author | Debi Oktia Haryeni Zata Yumni Awanis Martin Bača Andrea Semaničová-Feňovčíková |
| author_facet | Debi Oktia Haryeni Zata Yumni Awanis Martin Bača Andrea Semaničová-Feňovčíková |
| author_sort | Debi Oktia Haryeni |
| collection | DOAJ |
| description | A <i>k</i>-labeling from the vertex set of a simple 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> to a set of integers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></semantics></math></inline-formula> is defined to be a modular edge irregular if, for every couple of distinct edges, their modular edge weights are distinct. The modular edge weight is the remainder of the division of the sum of end vertex labels by modulo <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo></mrow></semantics></math></inline-formula>. The modular edge irregularity strength of a graph is known as the maximal vertex label <i>k</i>, minimized over all modular edge irregular <i>k</i>-labelings of the graph. In this paper we describe labeling schemes with symmetrical distribution of even and odd edge weights and investigate the existence of (modular) edge irregular labelings of joins of paths and cycles with isolated vertices. We estimate the bounds of the (modular) edge irregularity strength for the join graphs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi><mi>n</mi></msub><mo>+</mo><mover><msub><mi>K</mi><mi>m</mi></msub><mo>¯</mo></mover></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>n</mi></msub><mo>+</mo><mover><msub><mi>K</mi><mi>m</mi></msub><mo>¯</mo></mover></mrow></semantics></math></inline-formula> and determine the corresponding exact value of the (modular) edge irregularity strength for some fan graphs and wheel graphs in order to prove the sharpness of the presented bounds. |
| first_indexed | 2024-03-09T15:47:33Z |
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| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T15:47:33Z |
| publishDate | 2022-12-01 |
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| series | Symmetry |
| spelling | doaj.art-43ea0c0ec8e5454ba6dd18b4eaa39de92023-11-24T18:21:08ZengMDPI AGSymmetry2073-89942022-12-011412267110.3390/sym14122671Modular Version of Edge Irregularity Strength for Fan and Wheel GraphsDebi Oktia Haryeni0Zata Yumni Awanis1Martin Bača2Andrea Semaničová-Feňovčíková3Department of Mathematics, Faculty of Military Mathematics and Natural Sciences, The Republic of Indonesia Defense University, IPSC Area, Sentul, Bogor 16810, IndonesiaDepartment of Mathematics, Faculty of Mathematics and Natural Sciences, University of Mataram, Jalan Majapahit No. 62, Mataram 83125, IndonesiaDepartment of Applied Mathematics and Informatics, Technical University, 042 00 Košice, SlovakiaDepartment of Applied Mathematics and Informatics, Technical University, 042 00 Košice, SlovakiaA <i>k</i>-labeling from the vertex set of a simple 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> to a set of integers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></semantics></math></inline-formula> is defined to be a modular edge irregular if, for every couple of distinct edges, their modular edge weights are distinct. The modular edge weight is the remainder of the division of the sum of end vertex labels by modulo <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo></mrow></semantics></math></inline-formula>. The modular edge irregularity strength of a graph is known as the maximal vertex label <i>k</i>, minimized over all modular edge irregular <i>k</i>-labelings of the graph. In this paper we describe labeling schemes with symmetrical distribution of even and odd edge weights and investigate the existence of (modular) edge irregular labelings of joins of paths and cycles with isolated vertices. We estimate the bounds of the (modular) edge irregularity strength for the join graphs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>P</mi><mi>n</mi></msub><mo>+</mo><mover><msub><mi>K</mi><mi>m</mi></msub><mo>¯</mo></mover></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>n</mi></msub><mo>+</mo><mover><msub><mi>K</mi><mi>m</mi></msub><mo>¯</mo></mover></mrow></semantics></math></inline-formula> and determine the corresponding exact value of the (modular) edge irregularity strength for some fan graphs and wheel graphs in order to prove the sharpness of the presented bounds.https://www.mdpi.com/2073-8994/14/12/2671(modular) irregular labelingirregularity strength(modular) edge irregular labeling(modular) edge irregularity strengthwheelfan graph |
| spellingShingle | Debi Oktia Haryeni Zata Yumni Awanis Martin Bača Andrea Semaničová-Feňovčíková Modular Version of Edge Irregularity Strength for Fan and Wheel Graphs Symmetry (modular) irregular labeling irregularity strength (modular) edge irregular labeling (modular) edge irregularity strength wheel fan graph |
| title | Modular Version of Edge Irregularity Strength for Fan and Wheel Graphs |
| title_full | Modular Version of Edge Irregularity Strength for Fan and Wheel Graphs |
| title_fullStr | Modular Version of Edge Irregularity Strength for Fan and Wheel Graphs |
| title_full_unstemmed | Modular Version of Edge Irregularity Strength for Fan and Wheel Graphs |
| title_short | Modular Version of Edge Irregularity Strength for Fan and Wheel Graphs |
| title_sort | modular version of edge irregularity strength for fan and wheel graphs |
| topic | (modular) irregular labeling irregularity strength (modular) edge irregular labeling (modular) edge irregularity strength wheel fan graph |
| url | https://www.mdpi.com/2073-8994/14/12/2671 |
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