Two new topological indices based on graph adjacency matrix eigenvalues and eigenvectors

TL;DR

This paper introduces two novel topological indices, RV_a and RV_b, based on adjacency matrix eigenvalues and eigenvectors, reducing degeneracy issues in graph analysis, especially for alkanes and cyclic graphs.

Contribution

The paper proposes two new topological indices that incorporate eigenvectors alongside eigenvalues, improving discrimination among cospectral graphs compared to the Estrada index.

Findings

RV_a correlates with EE for certain alkanes

RV_b aligns with the Balaban index J

Indices show different sensitivities to graph structures

Abstract

The Estrada topological index EE, based on the eigenvalues of the adjacency matrix, is degenerate for cospectral graphs. By additionally considering the eigenvectors, two new topological indices are devised (RV_a and RV_b), which have reduced degeneracy for alkanes or cyclic graphs. Index RV_a shows similarity to EE in ordering of alkanes with 8 to 10 carbon atoms, whereas index RV_b is more similar to the average distance-based connectivity (Balaban index J). Inter-correlations between these four topological indices are discussed, indicating which factors have predominant influence.