On the Edge Metric Dimension of Certain Polyphenyl Chains release_5jdxu6edozdvbemjy7gemm4ahe

by Muhammad Ahsan, Zohaib Zahid, Dalal Alrowaili, Aiyared Iampan, imran Siddique, Sohail Zafar

Published in Journal of Chemistry by Hindawi Limited.

2021   Volume 2021, p1-6

Abstract

The most productive application of graph theory in chemistry is the representation of molecules by the graphs, where vertices and edges of graphs are the atoms and valence bonds between a pair of atoms, respectively. For a vertex <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>w</mi> </math> </jats:inline-formula> and an edge <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>f</mi> <mo>=</mo> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </math> </jats:inline-formula> of a connected graph <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mi>G</mi> </math> </jats:inline-formula>, the minimum number from distances of <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <mi>w</mi> </math> </jats:inline-formula> with <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </math> </jats:inline-formula> and <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6"> <msub> <mrow> <mi>c</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </math> </jats:inline-formula> is called the distance between <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7"> <mi>w</mi> </math> </jats:inline-formula> and <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8"> <mi>f</mi> </math> </jats:inline-formula>. If for every two distinct edges <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M9"> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mi>E</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>G</mi> </mrow> </mfenced> </math> </jats:inline-formula>, there always exists <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M10"> <msub> <mrow> <mi>w</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <msub> <mrow> <mi>W</mi> </mrow> <mrow> <mi>E</mi> </mrow> </msub> <mo>⊆</mo> <mi>V</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>G</mi> </mrow> </mfenced> </math> </jats:inline-formula> such that <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M11"> <mi>d</mi> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>w</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mfenced> <mo>≠</mo> <mi>d</mi> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mrow> <mi>f</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>w</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mfenced> </math> </jats:inline-formula>, then <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M12"> <msub> <mrow> <mi>W</mi> </mrow> <mrow> <mi>E</mi> </mrow> </msub> </math> </jats:inline-formula> is named as an edge metric generator. The minimum number of vertices in <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M13"> <msub> <mrow> <mi>W</mi> </mrow> <mrow> <mi>E</mi> </mrow> </msub> </math> </jats:inline-formula> is known as the edge metric dimension of <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M14"> <mi>G</mi> </math> </jats:inline-formula>. In this paper, we calculate the edge metric dimension of ortho-polyphenyl chain graph <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M15"> <msub> <mrow> <mi>O</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </math> </jats:inline-formula>, meta-polyphenyl chain graph <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M16"> <msub> <mrow> <mi>M</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </math> </jats:inline-formula>, and the linear [n]-tetracene graph <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M17"> <mi>T</mi> <mfenced open="[" close="]" separators="|"> <mrow> <mi>n</mi> </mrow> </mfenced> </math> </jats:inline-formula> and also find the edge metric dimension of para-polyphenyl chain graph <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M18"> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </math> </jats:inline-formula>. It has been proved that the edge metric dimension of <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M19"> <msub> <mrow> <mi>O</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </math> </jats:inline-formula>, <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M20"> <msub> <mrow> <mi>M</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </math> </jats:inline-formula>, and <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M21"> <mi>T</mi> <mfenced open="[" close="]" separators="|"> <mrow> <mi>n</mi> </mrow> </mfenced> </math> </jats:inline-formula> is bounded, while <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M22"> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> </math> </jats:inline-formula> is unbounded.
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