ScholarWorks@성균관대학교

초록

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 EF1(G)$E{F}_1(G)$, is defined as EF1(G)=& sum;vj is an element of V(G)edj3,$\begin{array}{r}E{F}_1(G)=\sum _{v_j\in V(G)}e{<}^{>}{d}_j{<}^{>}3,\end{array}$while the second version, denoted by EF2(G)$E{F}_2(G)$, is defined as EF2(G)=& sum;vivj is an element of E(G)edi2+dj2,$\begin{array}{r}E{F}_2(G)=\sum _{v_i{v}_j\in E(G)}e{<}^{>}{d}_i{<}^{>}2+d_j{<}^{>}2,\end{array}$where dj$d_j$ represents the degree of the vertex vj$v_j$ in G. We determine extremal graphs for both EF1$E{F}_1$ and EF2$E{F}_2$ 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.

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