First Hyper Zagreb Spectral Radii of Splitting and Shadow Graphs

The spectral radius RS of graph G is a spectral invariant derived from the eigenvalues of the associated matrix for a graph G. It is widely used in fields such as computer science, chemistry, biology, and network analysis. A notable expansion of this notion is the First Hyper Zagreb spectral radius, RSHZ1(G), which is defined as the largest absolute eigenvalues of the First Hyper Zagreb matrix. This study examines the behaviour of First Hyper Zagreb spectral radius under two graph operations, generalized shadow and generalized splitting. We compare RSHZ1(Splq(G)) and RSHZ1(Shq(G)) with the RSHZ1(G) of base graph G. Our study explores the impact of graph operations on spectral features, providing structural insights and analytical connections. The findings enhance our understanding of spectral graph theory and have potential implications in molecular computing and complicated network research, where graph operations are commonly used.