Sharp Bounds of First Zagreb Coindex for 
 F
 -Sum Graphs

<jats:p>Let <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M2"> <mi>G</mi> <mo>=</mo> <mfenced open="(" close=")" separators="|"> <mrow> <mi>V</mi> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi>E</mi> </mrow> </mfenced> </mrow> <mo>,</mo> <mi>E</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>G</mi> </mrow> </mfenced> </mrow> </mfenced> </math> </jats:inline-formula> be a connected graph with vertex set <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M3"> <mi>V</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>G</mi> </mrow> </mfenced> </math> </jats:inline-formula> and edge set <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M4"> <mi>E</mi> <mfenced open="(" close=")" separators="|"> <mrow> <mi>G</mi> </mrow> </mfenced> </math> </jats:inline-formula>. For a graph G, the graphs S(G), R(G), Q(G), and T(G) are obtained by applying the four subdivisions related operations S, R, Q, and T, respectively. Further, for two connected graphs <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M5"> <msub> <mrow> <mi>G</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>G</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </math> </jats:inline-formula>, <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M7"> <msub> <mrow> <mi>G</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>F</mi> </mrow> </msub> <msub> <mrow> <mi>G</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </math> </jats:inline-formula> are <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M8"> <mi>F</mi> </math> </jats:inline-formula>-sum graphs which are constructed with the help of Cartesian product of <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M9"> <mi>F</mi> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mrow> <mi>G</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </mfenced> </math> </jats:inline-formula> and <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M10"> <msub> <mrow> <mi>G</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </math> </jats:inline-formula>, where <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M11"> <mi>F</mi> <mo>∈</mo> <mfenced open="{" close="}" separators="|"> <mrow> <mi>S</mi> <mo>,</mo> <mi>R</mi> <mo>,</mo> <mi>Q</mi> <mo>,</mo> <mi>T</mi> </mrow> </mfenced> </math> </jats:inline-formula>. In this paper, we compute the lower and upper bounds for the first Zagreb coindex of these <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M12"> <mi>F</mi> </math> </jats:inline-formula>-sum (S-sum, R-sum, Q-sum, and T-sum) graphs in the form of the first Zagreb indices and coincides of their basic graphs. At the end, we use linear regression modeling to find the best correlation among the obtained results for the thirteen physicochemical properties of the molecular structures such as boiling point, density, heat capacity at constant pressure, entropy, heat capacity at constant time, enthalpy of vaporization, acentric factor, standard enthalpy of vaporization, enthalpy of formation, octanol-water partition coefficient, standard enthalpy of formation, total surface area, and molar volume. </jats:p>