Application of Resolvability Technique to Investigate the Different Polyphenyl Structures for Polymer Industry
Muhammad Faisal Nadeem, Mohsan Hassan, Muhammad Azeem, Salah Ud‐Din Khan, Mohammed Rafi Shaik, Mohammed El‐Meligy, Abdelatty Abdelgawad, Emad Mahrous Awwad
Abstract
Polyphenyl is used in a variety of applications including high-vacuum devices, optics, and electronics, and in high-temperature and radiation-resistant fluids and greases, it has low volatility, ionizing radiation stability, and high thermal-oxidative properties. The structure of polyphenyls can be represented using a molecular graph, where atoms represent vertices and bonds between atom edges. In a chemical structure, an item/vertex <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>v</a:mi> </a:math> resolves two items <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:msub> <c:mrow> <c:mi>v</c:mi> </c:mrow> <c:mrow> <c:mn>1</c:mn> </c:mrow> </c:msub> <c:mtext> and </c:mtext> <c:msub> <c:mrow> <c:mi>v</c:mi> </c:mrow> <c:mrow> <c:mn>2</c:mn> </c:mrow> </c:msub> </c:math> if <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M3"> <e:mtext>d</e:mtext> <e:mfenced open="(" close=")" separators="|"> <e:mrow> <e:msub> <e:mrow> <e:mi>v</e:mi> </e:mrow> <e:mrow> <e:mn>1</e:mn> </e:mrow> </e:msub> <e:mo>,</e:mo> <e:mi>v</e:mi> </e:mrow> </e:mfenced> <e:mo>≠</e:mo> <e:mtext>d</e:mtext> <e:mfenced open="(" close=")" separators="|"> <e:mrow> <e:msub> <e:mrow> <e:mi>v</e:mi> </e:mrow> <e:mrow> <e:mn>2</e:mn> </e:mrow> </e:msub> <e:mo>,</e:mo> <e:mi>v</e:mi> </e:mrow> </e:mfenced> </e:math> ; similarly, the ordered subset <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M4"> <m:mi>ϕ</m:mi> </m:math> of vertices resolves each pair of distinct vertices named as the resolving set, and its minimum cardinality is described as metric dimension. In the pharmaceutical industry, the competition to find new chemical entities for treating a disease dictates larger project teams that encompass more extensive and diverse synthetic efforts directed at increasingly complicated activity spectra. In this paper, we determine the metric dimension of para-, meta-, and ortho-polyphenyl structures, which are used for structure-activity analysis of these polyphenyl structures.