Second Hyper Zagreb Spectral Radii of Graph Operations

Abstract

Chemical graph theory is essential for deriving graph spectral radius, especially in quantum chemistry; a significant link exists between eigenvalues and this invariant of spectral graph theory. This invariant is derived from spectrum, and has numerous applications in various fields and is used to solve many real-life problems. Zagreb-type indices and their spectrum variants are crucial for QSAR/QSPR modeling, chemoinformatics, and computational drug design [16]. The literature contains numerous variants of graph spectral radius. In this article, we concentrate on a novel extension of the spectral invariant called Second Hyper Zagreb spectral radius, which is obtained by replacing the adjacency matrix with the Second Hyper Zagreb matrix. The two graph operations that are primarily covered in this article are splitting and shadow graphs. The connection between the Second Hyper Zagreb spectral radius of these graph operations and Second Hyper Zagreb spectral radius of base graph Gis of special relevance to us. By using these operations, we can address the challenge of figuring out the relationship between the spectral radius of the base graph Gand the spectral radius of the newly generated graph. The findings of this article improve our understanding and pave the way for future research in spectral graph theory, where graph operations are frequently employed and play an important role in solving real-world problems.