In mathematical chemistry and graph theory, a topological index is a numerical descriptor representing the structure of a molecule. The ability to distinguish between the structures of different molecules is crucial for the effectiveness of such indices. To enhance structural differentiation, researchers have proposed exponential versions of vertex-degree-based topological indices. It is known that bond incident degree indices generally possess a single exponential version, whereas vertex degree function indices—such as the forgotten index—give rise to two distinct exponential variants. Motivated by this observation, we introduce both exponential versions of the forgotten index for a graph G, with the aim of examining whether their mathematical properties and chemical behaviors align or differ. The first version, denoted by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$EF_1(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <msub> <mi>F</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, is defined as <Equation ID="Equ28"> <EquationSource Format="TEX">$\begin{aligned} EF_1(G)=\displaystyle {\sum \limits _{v_{j} \in V(G)}e^{d_j^3}}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mi>E</mi> <msub> <mi>F</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <munder> <mo movablelimits="false">∑</mo> <mrow> <msub> <mi>v</mi> <mi>j</mi> </msub> <mo>∈</mo> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </munder> <msup> <mi>e</mi> <msubsup> <mi>d</mi> <mi>j</mi> <mn>3</mn> </msubsup> </msup> </mrow> <mo>,</mo> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>while the second version, denoted by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$EF_2(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <msub> <mi>F</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, is defined as <Equation ID="Equ29"> <EquationSource Format="TEX">$\begin{aligned} EF_2(G)=\displaystyle {\sum \limits _{v_{i}\,v_{j} \in E(G)}e^{d_i^2+d_j^2}}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mi>E</mi> <msub> <mi>F</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <munder> <mo movablelimits="false">∑</mo> <mrow> <msub> <mi>v</mi> <mi>i</mi> </msub> <mspace width="0.166667em" /> <msub> <mi>v</mi> <mi>j</mi> </msub> <mo>∈</mo> <mi>E</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </munder> <msup> <mi>e</mi> <mrow> <msubsup> <mi>d</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>d</mi> <mi>j</mi> <mn>2</mn> </msubsup> </mrow> </msup> </mrow> <mo>,</mo> </mrow> </mstyle> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$d_j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> represents the degree of the vertex <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$v_j$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>v</mi> <mi>j</mi> </msub> </math></EquationSource> </InlineEquation> in G. We determine extremal graphs for both <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$EF_1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <msub> <mi>F</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$EF_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <msub> <mi>F</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> in several graph families, including trees, quasi-trees, unicyclic graphs, bicyclic graphs, trees with a fixed number of pendant vertices, and connected graphs with the same property. We also explore the chemical applicability of these indices and compare their effectiveness in chemical studies.

Exponential forgotten index and its application

期刊论文
发表日期：

2026年3月24日

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Kinkar Chandra Das,
Jayanta Bera

摘要

In mathematical chemistry and graph theory, a topological index is a numerical descriptor representing the structure of a molecule. The ability to distinguish between the structures of different molecules is crucial for the effectiveness of such indices. To enhance structural differentiation, researchers have proposed exponential versions of vertex-degree-based topological indices. It is known that bond incident degree indices generally possess a single exponential version, whereas vertex degree function indices—such as the forgotten index—give rise to two distinct exponential variants. Motivated by this observation, we introduce both exponential versions of the forgotten index for a graph G, with the aim of examining whether their mathematical properties and chemical behaviors align or differ. The first version, denoted by $EF_1(G)$ $E F_{1} (G)$ , is defined as $\begin{aligned} EF_1(G)=\displaystyle {\sum \limits _{v_{j} \in V(G)}e^{d_j^3}}, \end{aligned}$ $\begin{matrix} E F_{1} (G) = \sum_{v_{j} \in V (G)} e^{d_{j}^{3}}, \end{matrix}$ while the second version, denoted by $EF_2(G)$ $E F_{2} (G)$ , is defined as $\begin{aligned} EF_2(G)=\displaystyle {\sum \limits _{v_{i}\,v_{j} \in E(G)}e^{d_i^2+d_j^2}}, \end{aligned}$ $\begin{matrix} E F_{2} (G) = \sum_{v_{i} v_{j} \in E (G)} e^{d_{i}^{2} + d_{j}^{2}}, \end{matrix}$ where $d_j$ $d_{j}$ represents the degree of the vertex $v_j$ $v_{j}$ in G. We determine extremal graphs for both $EF_1$ $E F_{1}$ and $EF_2$ $E F_{2}$ in several graph families, including trees, quasi-trees, unicyclic graphs, bicyclic graphs, trees with a fixed number of pendant vertices, and connected graphs with the same property. We also explore the chemical applicability of these indices and compare their effectiveness in chemical studies.