Characterization of Extremal Graphs for Geometric-Arithmetic Index with Given Cut Vertices and Correlation with Eigenvalues

Abstract

This paper links spectral and chemical graph theories by examining the geometric-arithmetic and normalized Laplacian indices via Rayleigh's quotient principle. Spectral graph theory elucidates structural properties through the examination of eigenvalues and eigenvectors, whereas chemical graph theory represents molecular structures, presenting applications in chemistry. We concentrate on the collection $\mathbb{G}_{n}^{k}$, which includes all $n$-vertex graphs exhibiting connectivity 1 and containing precisely $k$ cut vertices, such that the elimination of a cut vertex results in the disconnection of the graph. Precise limits for the geometric-arithmetic index in $\mathbb{G}_{n}^{k}$ are established through auxiliary graph operations. Additionally, we enhance these bounds by analyzing the relationship between the normalized Laplacian index and the geometric-arithmetic index via the Rayleigh quotient, thereby offering a more profound understanding of the structural interaction of indices within these graphs