Novel eigenvector centrality indices for octane isomers to explore their physicochemical properties

Abstract

Abstract In chemical graph theory, a molecular structure is represented as a molecular graph $$G(V,E)$$ , where $$V$$ denotes the non-empty set of atoms (vertices) and $$E$$ represents the set of bonds (edges) between the atoms. Centrality measures in a molecular graph are vital for understanding the importance of individual atoms. Among various centrality measures, the eigenvector centrality is a robust metric that captures both the quantity and quality of connections to identify the most influential atoms. Mathematically, the eigenvector centrality $$({x}_{i})$$ of an atom $$i$$ in $$G(V,E)$$ can be defined as the $${i}^{th}$$ entry in the normalized eigenvector corresponding to the largest eigenvalue $$(\lambda )$$ of the adjacency matrix $$A\left(G\right)=\left({a}_{ij}\right)$$ , where $${a}_{ij}=1$$ if an atom $$i$$ is adjacent to an atom $$j$$ and $${a}_{ij}=0$$ otherwise. That is, $${x}_{i}=\frac{1}{\lambda }\sum_{j=1}^{n}{a}_{ij}{x}_{j}$$ where $$n$$ is the number of atoms in $$G(V,E)$$ . In this paper, seven eigenvector centrality-based topological indices are introduced and applied to octane isomers. These indices are utilized in QSPR (Quantitative Structure–Property Relationship) analysis to investigate the properties such as density, mean radius, entropy and more. The results establish a statistically significant and strong correlation between the computed indices and properties of octane isomers. The reliability and accuracy of the regression models are further confirmed through Y-randomization and chi-square goodness-of-fit tests, highlighting the potential of these indices for applications in cheminformatics-based predictive modeling.