A Spectral Approach to Join Based Operations on Graphs

The success of Spectral Graph Theory is due to its immense applications in several fields of science and technology. The parameter `graph energy' plays a significant role in the Hückel Molecular Orbital Theory, where it is used to approximate the total π -electron energy of conjugated hydrocarbons. Several modified versions of the energy parameter have been defined by many researchers depending on the context. This study explores the spectral characteristics and energy distributions associated with selected graph operations derived from the first Zagreb, second Zagreb, and sum-connectivity matrices. When the adjacency matrix provides information only on the existence of a relation among vertex pairs, these refined matrices give more information and weightage to the relation. These matrix-based energies quantify the total spectral energy of a system and provide insight into how structural modifications affect global stability and information distribution. The characteristic polynomials and corresponding eigenvalue spectra of the central graph are obtained for each matrix type, revealing how the spectral measures evolve under structural transformation. The analysis is further extended to the Indu–Bala product of graphs, a non-commutative operation that intricately merges the structural components of two graphs. The explicit expressions for the characteristic polynomial and spectral energies are derived, illustrating the effect of graph composition on spectral distribution. Additionally, for the operations of double-duplication and duplication vertex-join, the closed-form relations for the Zagreb and sum-connectivity spectra and their corresponding energy values are presented. The obtained results contribute to understanding how algebraic operations induce spectral energy shifts analogous to perturbations in physical or molecular graph systems.