Forgotten Coindex for the Derived Sum Graphs under Cartesian Product

A topological index (TI) is a molecular descriptor that is applied on a chemical structure to compute the associated numerical value which measures volume, density, boiling point, melting point, surface tension, or solubility of this structure. It is an efficient mathematical method in avoiding laboratory experiments and time-consuming. The forgotten coindex of a structure or (molecular) graph <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>H</a:mi> </a:math> is defined as the sum of the degrees of all the possible pairs of nonadjacent vertices in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:mi>H</c:mi> </c:math> . For <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M3"> <e:mi>D</e:mi> <e:mo>∈</e:mo> <e:mfenced open="{" close="}" separators="|"> <e:mrow> <e:mi>S</e:mi> <e:mo>,</e:mo> <e:mi>R</e:mi> <e:mo>,</e:mo> <e:mi>Q</e:mi> <e:mo>,</e:mo> <e:mi>T</e:mi> </e:mrow> </e:mfenced> </e:math> and the connected graph <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" id="M4"> <j:mi>H</j:mi> </j:math> , the derived graphs <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" id="M5"> <l:mi>D</l:mi> <l:mfenced open="(" close=")" separators="|"> <l:mrow> <l:mi>H</l:mi> </l:mrow> </l:mfenced> </l:math> are obtained by applying the operations <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M6"> <q:mi>S</q:mi> </q:math> (subdivided), <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" id="M7"> <s:mi>R</s:mi> </s:math> (triangle parallel), <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" id="M8"> <u:mi>Q</u:mi> </u:math> (line superposition), and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" id="M9"> <w:mi>T</w:mi> </w:math> (total graph), respectively. Moreover, a derived sum graph ( <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" id="M10"> <y:mi>D</y:mi> </y:math> -sum graph) is obtained by the Cartesian product of the graph <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M11"> <ab:msub> <ab:mrow> <ab:mi>H</ab:mi> </ab:mrow> <ab:mrow> <ab:mn>2</ab:mn> </ab:mrow> </ab:msub> </ab:math> with the graph <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" id="M12"> <cb:mi>D</cb:mi> <cb:mfenced open="(" close=")" separators="|"> <cb:mrow> <cb:msub> <cb:mrow> <cb:mi>H</cb:mi> </cb:mrow> <cb:mrow> <cb:mn>1</cb:mn> </cb:mrow> </cb:msub> </cb:mrow> </cb:mfenced> </cb:math> . In this study, we compute forgotten coindex of the <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" id="M13"> <hb:mi>D</hb:mi> </hb:math> -sum graphs <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" id="M14"> <jb:msub> <jb:mrow> <jb:mi>H</jb:mi> </jb:mrow> <jb:mrow> <jb:mn>1</jb:mn> <jb:mo>+</jb:mo> <jb:mi>S</jb:mi> </jb:mrow> </jb:msub> <jb:msub> <jb:mrow> <jb:mi>H</jb:mi> </jb:mrow> <jb:mrow> <jb:mn>2</jb:mn> </jb:mrow> </jb:msub> </jb:math> ( <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" id="M15"> <lb:mi>S</lb:mi> </lb:math> -sum), <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" id="M16"> <nb:msub> <nb:mrow> <nb:mi>H</nb:mi> </nb:mrow> <nb:mrow> <nb:mn>1</nb:mn> <nb:mo>+</nb:mo> <nb:mi>R</nb:mi> </nb:mrow> </nb:msub> <nb:msub> <nb:mrow> <nb:mi>H</nb:mi> </nb:mrow> <nb:mrow> <nb:mn>2</nb:mn> </nb:mrow> </nb:msub> </nb:math> ( <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" id="M17"> <pb:mi>R</pb:mi> </pb:math> -sum), <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" id="M18"> <rb:msub> <rb:mrow> <rb:mi>H</rb:mi> </rb:mrow> <rb:mrow> <rb:mn>1</rb:mn> <rb:mo>+</rb:mo> <rb:mi>Q</rb:mi> </rb:mrow> </rb:msub> <rb:msub> <rb:mrow> <rb:mi>H</rb:mi> </rb:mrow> <rb:mrow> <rb:mn>2</rb:mn> </rb:mrow> </rb:msub> </rb:math> ( <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" id="M19"> <tb:mi>Q</tb:mi> </tb:math> -sum), and <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" id="M20"> <vb:msub> <vb:mrow> <vb:mi>H</vb:mi> </vb:mrow> <vb:mrow> <vb:mn>1</vb:mn> <vb:mo>+</vb:mo> <vb:mi>T</vb:mi> </vb:mrow> </vb:msub> <vb:msub> <vb:mrow> <vb:mi>H</vb:mi> </vb:mrow> <vb:mrow> <vb:mn>2</vb:mn> </vb:mrow> </vb:msub> </vb:math> ( <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML" id="M21"> <xb:mi>T</xb:mi> </xb:math> -sum) in the form of various indices and coindices of the factor graphs <zb:math xmlns:zb="http://www.w3.org/1998/Math/MathML" id="M22"> <zb:msub>