| Summary: | Given any function <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>:</mo> <msup> <mi mathvariant="double-struck">Z</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msup> </mrow> </semantics> </math> </inline-formula>, let us define the <i>f</i>-index <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>I</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mo>∑</mo> <mrow> <mi>u</mi> <mo>∈</mo> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </msub> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>d</mi> <mi>u</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> and the <i>f</i>-polynomial <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>P</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mo>∑</mo> <mrow> <mi>u</mi> <mo>∈</mo> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </msub> <msup> <mi>x</mi> <mrow> <mn>1</mn> <mo>/</mo> <mi>f</mi> <mo>(</mo> <msub> <mi>d</mi> <mi>u</mi> </msub> <mo>)</mo> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>. In addition, we define <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>P</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mo movablelimits="true" form="prefix">lim</mo> <mrow> <mi>x</mi> <mo>→</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> </mrow> </msub> <msub> <mi>P</mi> <mi>f</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. We use the <i>f</i>-polynomial of a large family of topological indices in order to study mathematical relations of the inverse degree, the generalized first Zagreb, and the sum lordeg indices, among others. In this paper, using this <i>f</i>-polynomial, we obtain several properties of these indices of some classical graph operations that include corona product and join, line, and Mycielskian, among others.
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