Modular Sumset Labelling of Graphs
<p id="p1">Graph labelling is an assignment of labels or weights to the vertices and/or edges of a graph. For a ground set <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mi>X</mml:mi></mml:math></span> of integers, a sumset labelling of a graph is...
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| Autor principal: | |
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| Formato: | Capítulo de libro |
| Lenguaje: | inglés |
| Publicado: |
IntechOpen
2020
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| Materias: | |
| Acceso en línea: | https://www.intechopen.com/chapters/72952 |
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| Sumario: | <p id="p1">Graph labelling is an assignment of labels or weights to the vertices and/or edges of a graph. For a ground set <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mi>X</mml:mi></mml:math></span> of integers, a sumset labelling of a graph is an injective map <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mi>f</mml:mi><mml:mo>:</mml:mo><mml:mi>V</mml:mi><mml:mfenced open="(" close=")"><mml:mi>G</mml:mi></mml:mfenced><mml:mo>→</mml:mo><mml:mi mathvariant="script">P</mml:mi><mml:mfenced open="(" close=")"><mml:mi>X</mml:mi></mml:mfenced></mml:math></span> such that the induced function <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"><mml:msup><mml:mi>f</mml:mi><mml:mo>⊕</mml:mo></mml:msup><mml:mo>:</mml:mo><mml:mi>E</mml:mi><mml:mfenced open="(" close=")"><mml:mi>G</mml:mi></mml:mfenced><mml:mo>→</mml:mo><mml:mi mathvariant="script">P</mml:mi><mml:mfenced open="(" close=")"><mml:mi>X</mml:mi></mml:mfenced></mml:math></span> is defined by <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"><mml:msup><mml:mi>f</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mfenced open="(" close=")"><mml:mi mathvariant="italic">uv</mml:mi></mml:mfenced><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mfenced open="(" close=")"><mml:mi>u</mml:mi></mml:mfenced><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mfenced open="(" close=")"><mml:mi>v</mml:mi></mml:mfenced></mml:math></span>, for all <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m5"><mml:mi mathvariant="italic">uv</mml:mi><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mfenced open="(" close=")"><mml:mi>G</mml:mi></mml:mfenced></mml:math></span>, where <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m6"><mml:mi>f</mml:mi><mml:mfenced open="(" close=")"><mml:mi>u</mml:mi></mml:mfenced><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mfenced open="(" close=")"><mml:mi>v</mml:mi></mml:mfenced></mml:math></span> is the sumset of the set-label, the vertices <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m7"><mml:mi>u</mml:mi></mml:math></span> and <span class="inline-formula"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m8"><mml:mi>v</mml:mi></mml:math></span>. In this chapter, we discuss a special type of sumset labelling of a graph, called modular sumset labelling and its variations. We also discuss some interesting characteristics and structural properties of the graphs which admit these new types of graph labellings.</p> |
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| ISBN: | 978-1-83968-050-2 |
| DOI: | 10.5772/intechopen.92701 |